Why does Katniss take Buttercup back to District 13?
Have you ever heard the saying that we hate others when we see ourselves in them? This is absolutely true for Buttercup and Katniss, especially in Mockingjay, when they become more similar than ever. When Katniss is air lifted to District 13, everyone else from District 12 has already relocated and made their home there. Gale is getting in deep with the District 12 military and gaining rank and prestige as a resistance fighter, her mother and Prim are working with the other medical staff, even other Victors like BeeTee have a place in this new world. Katniss, however, does not fit in. She hides in closets and naps and ignores the schedule on her wrist. Her heart is still with District 12.
So when she returns and finds Buttercup there, living off rats among the blasted ruins, she sees a kindred spirit. As much as she hates the cat for its uselessness and competition for Prim's affection, she also can see that he is a stubborn survivor, just like she is. Though her stated reason for bringing Buttercup back is to make Prim happy, I think she also has a new-found respect for the creature.