Babbitty Rabbitty was a witch in the wizarding fairy tale “Babbitty Rabbitty and the Cackling Stump” from The Tales of Beedle the Bard. In this fable, Babbitty Rabbitty outwits a greedy king and his “charlatan” sorceror who try to round up wizards and witches so they can have all the magic to themselves (TBB/BR).
This name (and story title) may be a tongue-in-cheek reference to the books Rowling wrote when she was five or six years old: "The first finished book I did was a book called ‘Rabbit,’ um, about a Rabbit called Rabbit, thereby revealing the imaginative approach to names that has stood me in such good stead ever since. I wrote the Rabbit stories for ages to the point where it was a series of books about Rabbit which were very dull, illustrated by the author." (HPM)