There are many benefits for both the child and families in having a pet, however big or small.
Companionship - A pet can be a great friend for a child, especially if the child has a hard time making friends or communicating with people. Dogs are particularly sensitive and loyal.
Responsibility - Animals are a fun and easy way to teach your child how to care for a living thing and be responsible. These are just some of the skills your child will need for later in later to be successful adults.
Happiness - Pets can bring a lot of joy to a child and help a family have something to bond over. Ask your child what sort of pet they would like and why, although do not buy them a pet just because they ask for one. Make sure they understand how to look after their chosen pet.
Some children grow up having a pet and take great comfort in the memories that they have. However if your child is not used to having a pet of their own then perhaps start with something small. So if they go on and on about getting a puppy or a kitten then perhaps get them a hamster to start with and see how they get on with that.