Wow, a question I can actually answer!
Adrienne Mayor has written two books about the influence of fossils on mythology:
The First Fossil Hunters: Paleontology in Greek and Roman Times. Princeton University Press, 2000. rev ed 2010, new introduction.
and
Fossil Legends of the First Americans. Princeton University Press, 2005
It's been a while since I read these, so I recommend you pick them up yourself for more details.
My recollection is that there is some hard evidence for fossils in ancient civilizations, particularly in Greece and Rome, and then subsequently in Christian Europe after the fall of the Western Roman Empire. Beyond that, Mayor does quite a bit of speculation around the possibility of various mythological creatures being based on fossil finds, but with significantly less evidence.
Leaving the speculation aside, the Greeks and Romans did find various fossils, often the fossils of large mammals, such as mammoths or wooly rhinos, rather than necessarily dinosaurs. These bones were often held and displayed as relics, revered as pieces of mythical heroes or giants.
From the new intro to First Fossil Hunters:
Another dramatic moment came after months of searching in vain for what proved to be the fossil femur of an Ice Age woolly rhino, treasured by ancient Greeks and unearthed nearly two millennia later by archaeologists in the 1970s. Once stored on the ancient acropolis at Nichoria, southern Greece, perhaps revered as the thigh bone of a mythic giant, this fossil was presumed lost when I began writing The First Fossil Hunters. After a long, involved correspondence, to everyone’s surprise, the precious relic—a rare example of a large animal fossil collected in antiquity and recovered by modern archaeologists—finally turned up in a cellar in Duluth, Minnesota, where it had gathered dust for more than twenty years. The story of this historic bone is told in chapter 4. From its origins in the Peloponnese, the ancient fossil has traveled from the Mediterranean across the Atlantic to northern Minnesota, then east to New Jersey and New York City, west again to Bozeman, Montana, and Palo Alto, California. Now, ten years after the original publication of The First Fossil Hunters, I’m delighted to say that the peripatetic Nichoria Bone is en route to a permanent home in the Ashmolean Museum, Oxford University, where it will be displayed in the Greece Gallery.
It seems that this Greco-Roman tradition continued into Christian Europe, with various fossils being housed as relics, purported to be the bones of saints.
In China, "dragon bones" have been used as a component of traditional medicine for millennia, and are often fossils of ancient mammals.
As for some of the speculation, perhaps one of the most convincing examples is that the Greek myth of the Cyclops may have been inspired by ancient finds of fossilized dwarf elephant skulls, which are known from all over the Mediterranean.
There is much more in the books, I recommend reading them! I can't say that I'm 100% convinced by all of the speculative connections to mythology, but certainly ancient peoples did encounter fossils of extinct animals.
This is really similar to a question that I've answered before, so I'll repost it below!
So there is a common assumption that people found the fossils of ancient creatures and assumed them to be mythical figures. This after all makes a certain amount of logical sense, and it is not hard to see how the remains of an animal such as an elephant in Greece could become construed as a cyclops or other mythical beast! There are a number of popular stories around on the internet that follow through with this line of reasoning, often applying Chinese myths about dragons to the preponderance of "dragon bones" that were sold as cures for ailments with the rich fossil troves of China.
Only.... this is not a particularly satisfying explanation. There are a number of reasons for this, but I will let an actual paleontologist do the explaining here
http://markwitton-com.blogspot.com/2021/03/dinosaur-fossils-and-chinese-dragons.html
the tl;dr is that there are some parts of China that may have used ancient mammalian fossils as bones, there's little indication this was widespread, or even common, and no evidence of dinosaurs being connected to Chinese dragon myths.
However this does beg the question, what did ancient people think of fossils? This is a tricky proposition because ancient societies did not really have paleontology as a field of science. Indeed the roots of paleontology as a science belong to the late 18th century in particular. While there are some isolated instances of particularly common fossils such as ammonites which were believed to have been the remains of snakes. There were even attempts to "restore" the look of these deceased and desiccated serpents Given the prominence of ammonites in fossil deposits, and their relative ease of acquisition, its not unusual that they would feature prominently in pre-modern understandings of fossils, but that's due to a number of unique features that I'll discuss below.
Before that intial period of paleontological interest though..... it gets murky. Part of the issue is that actually recognizing bones as bones is not the easiest things to do. We are used to seeing dinosaur and other fossils in museums as part of entire reconstructions. But this is not how they are found. Indeed, most dinosaur skeletons found in the field are enormously fragmentary, and getting 50% of a full skeleton is outrageously rare. This applies even more so to many of the distinguishing features of many fossil species. Skulls in particular tend to be quite delicate and easily destroyed over the millennia. Indeed the most common surviving bones tend to be small fragments or teeth, and these are often not helpful in reconstructing a whole animal. Indeed think back to some of the more primitive renditions of dinosaurs that made them....well giant lizards That particular reconstruction dates to the mid 1820's, just at the beginning of paleontology as a distinct field. Even by the 1830's though there was a recognition that these bones, especially marine species of the Mesozoic, did not have modern analogues in a traditional sense. In part because of the excellent preservation biases of marine species in England, the Victorian period had relatively good ideas of what many old marines species looked like.....roughly
In short, unless you know what you're doing, it is rather tricky to actually find enough of an animal to fully reconstruct its life appearance. Nor are they found willy nilly in every part of the world. Fossils can only be found in rocks that are of a certain age and type, and these ancient formations are variously located and often in inhospitable locations. Nor are fully formed fossils usually just sticking out of the rock to be found by any passerby. Even "complete" skeletons are found over a wide area by dedicated teams working with modern science. The possibility of some Medieval monk or Ancient Architect just stumbling across a whole skeleton is....implausible. But an ammonite is slightly different admittedly. Due to their small size and robust construction, and plentiful deposition, ammonite fossils are distributed world wide, however they were usually attributed to some local species such as rams' horns being shed, snakes, or other slightly curly creatures. They were not understood as distinct species, but this is not really surprising. To accurately describe fossils and reconstruct the animals that they once were takes a tremendous amount of learning that simply was not available to people throughout history. Without understanding that species can go extinct, that the world is ancient, that animals back then were not the same as animals today.... that's a lot of things to actually know....
And even back then there was common recognition that these species were old. The age of the Earth was poorly understood at this time, but even non-Bible thumpers like Charles Lyell were not quite sure of the exact age of the Earth, though a date of several millions of years was not considered implausible. That these were once living creatures of a primordial world was well understood around scientific circles even in the beginning of paleontology. However the fragmentary nature of most finds made exact reconstructions impossible, and the process of what dinosaurs and other fossil species actually looked like is still an ongoing process with new discoveries being made.
You'll notice though that I am still staying in roughly the same time period.... and that's because we have precious little evidence that people before the early Victorian period really identified fossils as belonging to distinct entities that were ancient species. This is for several reasons, not the least of which is the lack of quality preserved fossils and the lack of knowledge on how to construct them into plausible species. A skill that eluded even many professional paleontologists of the time! (thinking of the time when one American paleontologist reconstructed a plesiosaur with its head on its tail....) Usually they just associate them with some still extant species such as snakes, with the occasional mystical property added on (though this too is also not unusual for the time, medieval and early modern people had....odd ideas about the properties of various naturally occurring items and substances).
