I wasn't able to find any specific quotes from JK addressing this, but I did come across a post on MuggleNet from a contributor named Leah which speculated regarding a Celtic legend she happened upon in a bookstore. She explains the story in length, but summarizes it in this way:
Now, there are obvious similarities between [the main character's] mare and thestrals, such as their remarkable traveling abilities, their appearance, and their association with abstract things such as destiny, death, and magic. Rowling’s thestrals were carnivorous, which brings me back to the bird, kestrel.
Kestrels are mentioned in Quidditch Through the Ages as the name of a team, denoting that Rowling is familiar with them. Kestrels are dark birds of prey, with white patches below their eyes. Perhaps Rowling combined these birds with the mare of doom? One thing I also caught was N’oun Doare’s education by a Druid and his return at 17. Druids were ancient scholars and magicians who are believed to have practiced rituals in such places as Stonehenge. He reminds me very much of a certain young wizard…
I do not know if this is the story where J.K. Rowling discovered her intriguing creature, but one is allowed to speculate, no?
This seems a potential jumping off point for some further investigation, but of course not a solid answer.