To rehome a pet is to find it a new owner and a new home. You hear this most from animal shelters and rescue groups: a dog or cat has been looked after there, and now someone is trying to place it with a family who can keep it. Though actually, people also rehome pets when they move, fall ill, or can no longer care for them — it is not only a shelter word. It is a kind way of talking. It points at a new home, not at throwing the animal away. You rehome animals, not sofas, and not people (for people, English prefers rehouse). Neutral, everyday English in charity and pet talk; not a stiff legal word.