In the region of North America I focus on, dinosaur fossils are exceptionally rare. So I'll hope you'll forgive me for expanding the scope of your question a bit to cover fossils in general.

I'll start with an old example. Indian Knoll is an Archaic site in the west-central Kentucky, dating to between 5000 and 4000 years ago. The most notable features at the site are large shell midden. Freshwater shellfish made up a large portion of the diet at the time, and all those shells had to go somewhere. As the shell mounds built up, they were later re-purposed into funeral mounds for more than a thousand individuals.

Of particular interest here is that fossils also show up in the mound. The most common are brachiopods, mostly from the Ordovician. These closely resemble more familiar shellfish although they belong to a different phylum entirely. They're common in the area, and it seems that the people of Indian Knoll had a habit of collecting them and moving them into the shell mounds. Why they would do this is less clear. Perhaps the shells held some special significance; perhaps some weary parents were just trying to quietly dispose of their kids' inconvenient collection of fossils.

Other fossils found at the site include fragments of Calamites (a tree-sized horsetail from the Carboniferous) and tapir teeth from the Pleistocene. The tapir teeth in particular were found in association with a human burial, and their placement may suggest they were included as part of a medicine bundle or something similar collection of personally significant tokens of ceremonial power.

Jumping forward in time, the first mostly complete skeleton of a mastodon known to Europeans as originally excavated by Native Americans and presented as a gift to the French. These fossils had come from what's now Big Bone Lick State Park, also in Kentucky. The Lenape have a rather dramatic story to explain how the salt lick became filled with the bones of these extinct animals:

The yahquawhee (mastodons) were created to be beasts of burden and to serve humanity, but they rebelled. So great was their size and strength that they sought to dominate all life and ruthlessly attacked all other species. Eventually humans and other animals form an alliance against the yahquawhee and a great battle is fought against them in the Ohio Valley. The Great Spirit descended to earth to observe the outcome. At first, the battle went disastrously for the allies. The yahquawhee were too strong and their hides too thick to be harmed. The blood for all the fallen allies pooled up around the seemingly victorious yahquawhee.

As the earth turned to bloody mud, the yahquawhees' own immense weight proved to be their undoing. Those that didn't drown in their own victory became so mired that they were easily overwhelmed by the surviving allies. Only the leader of the yahquawhee remained. He battled the Great Bear, the strongest warrior on the alliance's side, and won. At this point, the Great Spirit struck the last yahquawhee with a bolt of lightning. He didn't kill the yahquawhee, but did cause him to flee far to the northwest - never to be seen again.

​ Adding this to u/Reedstilt's great answer for another region: modern-day central America, specifically central Mexico in colonial times. While many dinosaur fossils have been found in Mexico I'm also not aware descriptions of those for the earlier colonial period I'm studying. But there are descriptions of what might have been femurs, or possibly extinct giant ground sloths like Eremotherium or Megalonyx. I'll focus on the first European description we have from one of those, where the native Tlaxcaltecs present fossil bones - described as "giants' bones" - to Hernan Cortés during his conquest campaigns in central Mexico in the early 16th c. - this is adapted from an earlier follow up answer to another answer of mine.

I can't really speak to the archaeology part of the question, but rather to perceptions tied to the "giants' bones" at the time. In "Fossil Legends of the First Americans" Adrienne Mayor deals with various American legends connected to fossils, including this one. Mayor describes various Aztec codices and Inca traditions as the oldest documented fossil legends. They often saw the remains of mammoths and other large creatures as mythical giant beings from earlier periods. Some of these findings have even been confirmed by paleontological fieldwork.

The specific bones mentioned above come from the Tlaxcaltecs, a Nahua group of the Valley of Mexico that had been an enemy of the Aztecs/Mexica in pre-Hispanic times. They then became very important allies of Cortés during his conquest campaigns. In the chronicle mentioned, the Spanish conquistador Díaz de Castillo gives us the first account of an fossil legend in the Americas (written in the later 16th c., although Díaz was an eyewitness to these events). The Tlaxcaltecs tried to impress Cortés by showing him a huge bone:

They said that their ancestors had told them, that in times past there had lived among them men and women of giant size with huge bones, and because they were very bad people of evil manners that they had fought with them and killed them, and those of them who remained died off. So that we could see how huge and tall these people had been they brought us a leg bone of one of them which was very thick and the height of a man of ordinary stature, and that was the bone from the hip to the knee. I measured myself against it and it was as tall as I am although I am of fair size.

The Tlaxcaltecs also brought other bones of similar sizes but they were not as well conserved and not entire. The conquistadors were astonished "to see these remains, and knew for certain there had been giants in that land." The Spaniards then took those bones and would send them on the first shipment from Veracruz as a gift to the Spanish emperor Charles V. after the fall of Tenochtitlan. Cortés even sent further search parties which brought more of these bones. There are a few things to unpack here: The obvious fascination of the Spaniards for this bone and importance attached to it; and the pre-Hispanic Nahua legends tied to such bones.

According to Mayor

[a]t that time in Europe, the fossils of very large, extinct animals were believed to be relics of giant humans who drowned in Noah's Deluge, early Christian saints, or famous personages from classical antiquity. Since Homeric and biblical ties, it was commonplace that people of past ages had been giants. By the 18th century [...] most European Americans no longer believed that giant human races still existed [...]. But in 1519, many in Europe were convinced that bizarre creatures populated the New World, and they expected explorers to discover Amazons, cannibals, weird monster--and living giants. The colossal femur that Cortés acquired as a marvel from Mexico would have been seen as proof that giant tribes had dwelled--and perhaps still lived--in the Americas [Mayor, 79]

So on the one hand the connection between giant bones and giant humans would have fit very well with traditional European imaginations. They also fit with views early modern Europeans had of the still very unknown Americas and their people as mystical, monstrous creatures with a varying number of limbs and heads. These built on such older descriptions, e.g. that medieval Europeans made of people to the "far east".

On the other hand, there are also pre-colonial Nahua legends of giant people that had lived before the Aztecs - as reflected in the Tlaxcaltec story recounted by Díaz de Castillo. Another version of this is recounted much later by the Acolhua chronicler Fernando de Alva Ixtlilxochitl, writing in the early 17th century but building on earlier indigenous sources. In his Nahua version of the world's creation he mentions such a race of giants that was in the end subjugated by "ulmecas y xicalancas [olmecs and xicalancas]". However, Alva Ixltilxochitl was already well-educated in European religions and history, so it's hard to say how much European influence is contained in his description. Similarly the Tlaxcatlec story of the giants comes down to us from Díaz and may be influenced by his own views.

In the end, the most probably answer seems to be that the bones given to the Spaniards and sent to Europe came from an ancient fossil, probably a femur according to Mayor. I should add that fossils of mammoths have since been found in the Basin of Mexico, with new ones found quite recently. The story connecting them to giant people went hand in hand with imaginations of both Europeans and Nahua, and reflects the importance attributed to them by both groups. Apparently Adrienne Mayor searched for these first bones in European museum to no avail, which does not mean they might not still be lying around in some museum's reserve.

Edit: Pinging u/retailguypdx for their related follow up; spelling

Thanks for the wonderful answers by u/Reedstilt and u/drylaw. In addition to the Lenape and Tlaxcaltec examples, there are many others throughout the ancient world in which people found fossils, often attributing them to be of giants or some ancient now-extinct giant animals, and of course then incorporating these precious objects into their world as relics or proof of their stories.

The correct attribution of fossils as either extinct animals or long-since destroyed beings from an earlier creation event, shows the ingenuity of native natural philosophy. It shows us too that our understanding that the layers of earth hide its history, commonly thought to be modern, is actually a belief which stretches back millennia and across continents. We now know the necessity and power which comes from carefully recording an archeological dig, but archeology as well has also been handed down to us through the millennia. While there is not a direct connection to the modern tradition, people on both sides of the Atlantic have invented this type of natural science.

In the Americas, Mississippians incorporated bands of colors into the building of mounds, for the Cahokians this pattern was yellow-black in repetition, with yellow always first. This was obviously done for some important reason, but why, we do not know. All it shows us is that the layering of the earth within mounds was recognized and important. Human sacrifices at the Emerald Site show that they left the sacrificial pit uncovered, only covering it after a rain or snow event; suggesting that they associated this layering with the actions of “natural” forces. Yet they would go further with this line of thinking: Mississippians would sometimes return to an old "closed" temple to dig an intrusive tunnel down. And once they reached the now-buried floor of the old temple, they'd place some heirloom pieces of ceramics as an offering. We do not know their names, but the priests (?) who conducted these excavations were doing a form of archeology. As Timothy Pauketat styles them, they were ethno-scientists.

Yet there are other cases in which we do know their names. I'm sure there have been other cases, but one of the more recent examples of native archeology I'm aware of was carried out by a Hopi man, Lesso, in the 1890's. He had been a helper on a western dig at Sikyatki in 1895. That town had supposedly been created by the ancestors of his own clan, The Coyote clan; but the western-led dig didn't answer that question. Lesso did not give up, he wanted to know, and to do that he needed to do an excavation himself. First, he obtained permission to dig into a kiva (ritual building) at Sikyatki from the eldest member of the Coyote clan at Oraibi. When he did, he found a mural in that kiva which he copied. This mural showed a Coyote clan story, and proved to him that his clan's stories about Sikyatki were true.

"Lesso carefully drew the figures on a board with charcoal, planted a paho [prayer stick/bundle] at the base of the mural, and sealed up the kiva again. He then returned to Oraibi with the charcoal sketch he had drawn, paying his debt to his religious society and his clan. He had found what he wanted. The kiva mural depicted the Coyote-Swallow race and confirmed tradition that Sikyatki was the legendary home of the Coyote Clan." - Book of the Hopi, Frank Waters and Oswald White Bear Fredericks

On the other side of the Atlantic, the Neo-Babylonians are the best known for their practices of ancient archeology, as there were a few history-obsessed kings particularly Nabonidus. He employed an archeological team, led by an ancient textual specialist, and every summer they would conduct seasonal digs to find the foundations of old temples so that they could be uncovered and restored. Reading their "reports" often sounds like the trials and tribulations of modern archeology:

"I [Nabonidus] sought to rebuild this temple; and in order to do so, I opened up the ground inside Agade [Akkad] and looked for the foundation...for three years I excavated in the trench of Nebuchadnezzer. I looked to the right and left, to the front and rear of the trench. [Then] a downpour occurred and made a gully. I said to them, "Dig a trench in this gully". They excavated in that gully and found the foundation of Eulmash."

Besides archeology, there are two examples of indigenous ethno-scientists making reconstructions of these ancient animals. Peter Faris, a rock art researcher, notes that at the Cub Creek site in Utah there’s a panel of huge lizards next to much smaller humans. Nearby, there is a huge theropod fossil track panel. The makers of the Cub Creek site were the Fremont archeological culture, and it was made sometime contemporary with the European Middle Ages. While the people in that technocomplex now exist as the Ute and Paiute tribal identies, today of course, they all recognize fossil tracks as what they are. We don't know if their Fremont ancestors also recognized fossil tracks as a type of ancient animal print, but as Peter Faris mentions, the Navajo people do recognize fossil prints as naasho'illbahitsho biikee' meaning "big lizard tracks". As Robert Bednarik notes in his recent book “Myths About Rock Art”, the Hopi call a dinosaur track site in Utah “tsidii nabitin” which means “bird tracks”, although they were made by Kwaatoko the man-waterbird. So perhaps, like the Navajo, the Fremont also recognized the Cub Creek trackway as made by ancient huge lizards, and then created their reconstruction of those creatures (with tiny humans) at that nearby rock art panel.

The second example of paleo reconstruction by an indigenous person is in southern Africa, as explained by Robert Bednarik in the book just previously mentioned:

"As surprising as it may seem, we do have three authentic depictions of dinosaurs in world rock art, all three quite probably created by the same San or Bantu-speaking artist. They were painted in black pigment at Mokhali Cave in Lesotho, southern Africa, apparently depicting an orithopod extinct for more than 65 million years (Ellenberger et al. 2005).

How is that possible? Together with them is a nearby reddish rock painting of one of the many fossil sauropod tracks found nearby; Ellenberger, a rock art recorder and ichnologist, has described some 58 rock slabs bearing such fossil footprints from the region, the nearest set of tracks being 3km away. Moreover, there is a dinosaur skeleton preserved in the sandstone wall near the eastern end of Mokhali Cave. So the logical explanation is that the artist who can be assumed to have been an expert tracker (as is very common among his people), observed such fossilized tracks carefully and tried to decode deduce from them the kind of animal that would have made them.

...The artist's reconstruction of the ornithopod is superb: not only did he deduce from the tracks that the creature walked on two legs ending in bird-like feet, he also predicted a body shape that is rather close to reconstructions based on extensive skeletal material. The deductive ability of this indigenous palaeontologist is utterly remarkable (cf. Lockley 1991, 1999).”