Choosing a family pet is by no means an easy task. Parents need to take into account a number of factors such as how friendly the pet will be, how costly the upkeep will be, how much maintenance is required.
It depends on the age of the child and how responsible they are. I'm a fan of hermit crabs as starter pets for children since they are easy to look after and not really long lived (it's sad when they die, but it does make them less of a commitment than a dog or a cat). Pet rats are good too, because they are cute and furry and live a couple of years (my longest lived one was six when she died) provided you lay down some ground rules about where they are allowed to roam. Personally I wouldn't want them wandering freely through the whole house chewing on furniture and electrical wiring.
This is a tricky ones. Dogs are friendly, energetic and always so glad to see you. Cats, on the other hand are therapeutic, and when they're lap cats, they calm you down. I'd prefer cats, they're charming.
I think guinea pigs and rabbits are a good choice. They are small enough to be handled, and while need looking after just like any other pet, they do not need 24/7 attention like a dog. Their life span is also just about right. Smaller pets like hamsters, rats, and mice don't live that long, and it can be upsetting to see them die so often. Dogs and cats require a much bigger commitment, as they are going to be a part of the family well into to a young child's teens. Guinea pigs and rabbits live for about the length of childhood, after which, the child can make a more informed decision about what they want for themselves.
I think it really depends on the nature of the child. If a child is rough, has a tendency to be cruel or looses interest quickly, I really don't think that they should be given a pet at all. Animals also have feelings and like us, they can be hurt, so if you bring a pet of any kind into your family, be sure that you have a lifetime commitment to it. Dogs are wonderful companions for children as they're playful, very affectionate and come in all sizes. Gentler children may enjoy caring for a guinea pig or rabbit. However, most animals are quite social, so it's always a good idea to get at least two, so that they can keep each other company when you're not around. Also, it's important that you can provide adequate shelter for pets. Like us, animals are sensitive to the weather so it's ideal if you can keep your pets indoors, at least some of the time.
I grew up with a cat. I think they are the best pet for a child because cats are slightly more independent and a child is not ready to take on the emotional responsibilities of an animal that is wholly dependent on the child.