The rescuer Miss Lin called her Butter because she is very sticky and loves to stay with people.  She was raised by some neigbors who let her stay in the garden, so you can say she has a home but not really.  They moved away after a while and little Butter does not even have leftovers to eat then.  Ms Lin feeds her a few times, and smart Butter somehow managed to find her home and stay outside her door.  Ok, finding Butter a home has become Ms Lin's mission.