Very little is mentioned about Bun-Bun, other than that it is Amy's special stuffed animal. It can be inferred that it is a rabbit. The only information given is that Sophie was describing a picture of she and her sister, saying, "the two of them proudly holding up Ella and Bun-Bun - their special stuffed animals - to the camera" (pg 101 hardcover, Nightfall). She was probably given Bun-Bun after Sophie's allergy attack at age 9.
In Legacy, Amy mentions that Sophie made up a song and "a little hoppy dance".