Figured out the Greek myth yet? (Imagine me cackling...ok, I'm probably too jet-lagged on the other side of the world to really cackle, but imagine.)
Bellerophon was one of the ancient Greek heroes; he's the guy who rose Pegasus and fought the chimera. There's a more complete summary of his background here.
My story doesn't revolve around Bellerophon. But he was the inspiration for my main character's name. And I had my main character ride a pegasus. And the monster chimera is a key part of the story. And the main baddie dies from lead (like how Bellerophon killed the chimera).
But the story's about how a young girl is tempted by magic and must chose between saving her teacher or taking a powerful magic—the kind of magic she's always dreamed of. That's got nothing to do with the Greek myth. In truth, I could have left the Greek stuff out and told the same story. I didn't want to, though. Think about the Percy Jackson series—Rick Riordan could have written Percy without the Greek stuff, and just made up an entirely new world and set of gods to fit the story. But how boring would that be?
The best take-offs of myths and fairy tales are not the ones that are direct re-tellings, but the ones that change our ideas of what the myth really means, or that use a fairy tale to add another dimension to the story. Inspiration and re-tellings should enhance a story, not shape one.