I have to buck the trend and go with a dog. I think every child who is able should have a dog. They not only make good companions, but they help protect your house (if you get a big enough one, I wouldn't bring a dog smaller than a Lab into the house)
BUT, they do require a lot of responsibility, that you will have to share with the child, as NO 10 year old will be able to be 100% responsible for any animal. They may forget to feed them, forget to clean their cages (and animals like rats, mice, rabbits, hamsters etc really smell when their cages aren't cleaned every day or so.) A Golden Retriever or a chocolate or black lab is a good "Starter Dog" because they make good pets and usually want to please their owners. But, don't get one unless you can dedicate at least a few hours a day to training, learning and care.
For the animals above, rabbits live the longest. Yes. But, the males smell terrible (we've had rabbits and have a male now, and he's sweet, but damn he smells) and the other animals require a lot of care and responsibility. We had a a hamster get out of his cage, fall into a vent and end up getting killed by the fan on the furnace. It was SO sad. Your child will have to make sure the animal is safe, has the right food in the proper quantities, is kept clean etc.
I wouldn't recommend a gerbil because they sleep all day and don't play until night. If they are kept in your child's room, they will keep her awake with playing, eating, drinking and squeaking all night.