Where did the name hillbilly originate?

It is my understanding that the was a family named Billy and they lived in Kentucky. As the family grew and spread out, some of the family lived on a nearby hill or mountain whereas others lived in the valley. When anyone queried as to where the Billlys were, the question was answered thusly: "Do you mean the hillBillys or the valleyBillys?"

In time the hillBillys developed an unsavory reputation and therefore the bad connotation.