Bun-Bun was first mentioned on Nightfall, while Amy was in the lost cities. Sophie and Amy were looking at photos in Sophie's old Album, and one of the photos showed the two of them holding up Ella and Bun-Bun to the camera.
On Legacy, when Sophie and Mr. Forkle visited Amy's house in order to bring back their missing hard memory from their childhood, Amy showed Bun-Bun as an evidence that she has mainly good memories from their time together. Amy explained that she likes Bun-Bun because of how Sophie was entertaining her with Ella and Bun-Bun when they were little. She mimics the voice the Sophie used when they were little and even mentioned a dance. Amy kept it during the moves and identity changes of her family, even though all the other stuffed animals were replaced. It looked somewhat old, and Sophie found that to make it perfect since it looked original. The whole conversation helped Sophie to accept that Amy wants to get the hard memory back too, and that she is not going to hate Sophie.