Very little is mentioned about Bun-Bun, other than that it is Amy's special stuffed animal. 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 possibly given Bun-Bun after Sophie's allergy attack at age 9. As of Legacy, it is not as white as it used to be, and it is matted in some areas.
In Legacy, Amy mentions that Sophie made up a song and "a little hoppy dance". We also know that Bun-Bun is Amy’s favorite stuffed animal. It is her favorite because Sophie used to make Bun-Bun talk to Amy with a high-pitched voice. She may have said ”Hey there Miss Amy, would you like to play?” This happened when Sophie was 7 and Amy was 4.