This is something I've wondered about for a long time. Dinosaur fossils have (obviously) existed for the entire history of humanity, and while some are deeply buried, many (especially the first ones discovered) are near or on the surface, and quite easy to find. This, combined with the fact that humans have been digging or mining for thousands of years -- albeit at a much smaller scale than the last century or two -- suggests that these fossils must have been discovered pretty frequently by ancient people.
Since the concept of evolution, and our current understanding of the age of the earth, are recent developments, what would a person discovering a dinosaur fossil 1,000 or 2,000 years ago have thought of them? They would obviously have to find some way to fit this bizarre creature into their understanding of the world, and it would be reasonable to conclude that it represented an animal that might be still alive, even if nobody has ever seen it.
And if somebody 2,000 years ago imagined that a mysterious, giant, possibly reptilian animal was lurking somewhere out in the wilderness, it stands to reason that that imagined creature would end up looking a bit like the dragons of folklore.
Is there any evidence to suggest that this is a possible origin of dragon myths, or is it just wild speculation on my part?
It is possible, yes, that fossils (not necessarily just dinosaur fossils) did play a role in the development of dragon myths. It's mostly all speculation, really.
Most fossils that people find are either recognizably some smallish organism, like a seashell or a small fish, or a disarticulated bone or piece of a bone. All perfectly laid-out skeletons just don't really happen very often at all, and even then it takes a keen eye to discern what is what.
We do know that prior to Georges Cuvier's pioneering work in comparative anatomy, people very often tended to assume large fossil bones were the bones of creatures that were still alive at the time, or possibly giant versions of them. In the Western world, it was also frequently assumed they were the remains of animals and people killed in Noah's flood. One particular thigh bone (now known to be from a type of meat-eating dinosaur) was originally thought to have been the thigh bone from a giant antediluvian person, then for the while the fossilized scrotum of said person, then recognized again for a leg and not a ballsack, and finally correctly identified as not from a flood-victim giant human at all.
Also, prior to Cuvier's work, we really didn't have a concept of extinction. The world at this point was also not fully-explored by any one group, so there was an idea that even though these giant critters may not be around here anymore, the might still be found out there somewhere. Thomas Jefferson actually gave specific instructions to Lewis and Clark for their westward exploration to try and find a living specimen of a giant ground sloth--an animal known from fossils and which the founding father assumed based on assumptions that were perfectly rational for the time to still be kicking around out there somewhere.
There are some speculations that specific fossils may have influenced certain mythological creatures. For example, some myths of the Griffin describe a four-legged creature with wings, a beak, a horn on the back of the head, and lives in the desert and guards gold. The dinosaur Protoceratops has four legs, shoulder blades that to the untrained can look sort of wing-like, a beak, a frill on the back of the head which if the sides break off--which is very common--can look like a horn, its remains are found in the deserts of Mongolia and China in areas where gold deposits are not unheard of. It's possible, maybe even plausible, but not really testable.
Stories of mythical beasts, like dragons, grow and change with the times. They ultimately have many sources, some from the natural world, some purely fantasy. Fossils may explain part of the stories, but not all of them.
Consider that if you go to the Roman and Medieval sources, a "dragon" is described many different ways, many of them only having a passing resemblance to what we today would agree on is a prototypical "dragon." For example, in some medieval bestiaries based on Pliny the Elder's works, dragons are described as giant snakes that live in the East (i.e. India), live in trees and hunt elephants by dropping on them from the trees and constricting them, and are deathly afraid of jaguars.
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I wrote about this recently, so I'll just copy and paste my earlier answer. The people of Mesoamerica had discovered dinosaur bones, but they thought that the bones were the bones of giants. This account is from Bernal Diaz, a soldier who fought with Cortes in conquering the Aztecs: "The Aztecs said that their ancestors told them that a long time ago there were settlements of men and women who were very tall with huge bones; that because they were very evil and of bad behavior, they killed them fighting with them, and those remaining died. And in order that we could see what enormous bodies they had, they brought a leg bone of one of them, and it was very heavy and very large in height compared to a man of average stature. And that leg bone was from the knee to the hip. I measured myself with it and it was as tall as I, although I am of average height. And they brought other bones like the first. Most were already bare and hardened from the earth, and all of us marveled at seeing those leg bones, and we were certain that there had been giants in this country. And our Captain Cortes told us that he would be sending that great bone to Spain in order that His Majesty could see it, and so thus we sent it with the first products that went out."
Hi! Is it OK to expand on a subject, providing (sourced) additional information? While I cannot add to the current answers about specifically dinosaur fossils, I have a bit of knowledge that I imagine might be of interest to the /u/diamond and others here. I'll post the comment, and hope I do not break the rules... if I do, mods, my apologies.
I would like to add something on the subject of ancient people interacting with fossils, altought much younger than dinosaurs; namely, fossilized mammoths in Siberia. These animals went extinct in the end of the last Ice Age (more exactly, of the last glaciation), about 20,000 to 13,000 years ago (let's ignore for now the dwarf mammoths of those remote islands in northern Siberia that managed to survive until much later, in fact they were some still alive at the time when early Pharaohs ruled Egypt^1). Moreover, the mammoths were prone to die in bogs or in regions where the permafrost (frozen underground) helped preserve their remains. As a result, there were (and still are) somewhat frequent discoveries of fossilized mammoth bones in Siberia.
What did ancient people think of them? As for nomadic people native to Siberia, it is hard to say because there is little to no written record of their histories and myths (to my knowledge). We do, however, possess a very fascinating (to my opinion) written record by the Arab traveler Ibn Fadlan. The man was a merchant and an envoy of the Caliph who traveled north from Baghdad in the lands of the Bulghars, the Khazars, the Rus, and many Turkic tribes in the steppes of western Siberia / eastern Europe. I encourage you to read the book: Ibn Fadlan and the Land of Darkness: Arab Travellers in the Far North^2.
Sorry, I'll get to the point now. Ibn Fadlan recalls being shows bone-like objects that a man claims come from a Rhinoceros. While he (Fadlan) accepts this as good enough of a fact, we now know there weren't any rhinos in this part of Siberia that the bone could've come from. In the Penguin edition of the book (see below), Caroline Stone and Paul Lunde comment that the material could have come from fossilized mammoth tusk...
The three plates of onyx-like material (...) may have been made of the material called khutu, fossilized mammoth tusk; this was much sought after to make knife handles, because of its durability.
1: Vartanyan, S. L.; Garutt, V. E.; Sher, A. V. (1993). "Holocene dwarf mammoths from Wrangel Island in the Siberian Arctic". Nature 362 (6418): 337–349 -- yes, the precise source was fetched from this simpler wikipedia article
2: https://www.penguin.co.uk/books/187609/ibn-fadlan-and-the-land-of-darkness/