When you adopt children in Skyrim: Hearthfire, after they move in to live with you, there will be a chance every day that they will return home with a pet, They will ask you whether or not you will let them keep it. If you agree, they will name it and the pet remains in your house to play with the kids.
Number of pets you can have
Your children can have up to two pets at a time. If one pet dies or is killed, the children will be upset for a few days but then forgive you and get another pet.
List of Pets
The following are the list of possible pets your child brings home:
Dwarven Automaton Spider