I'm slightly allergic to some cats, which can be a pain as i am a vet nurse!! I've found burmese to be the worst. Rex cats, as mentioned by Califrau, maybe aren't the cutest, but i have met some really friendly ones. Also, IME rabbits can be just as bad, as they have fluffy undercoats and can moult like mad. Also need to be handled a LOT to become friendly, are prone to being ill quite a lot, and can scratch & bite.
Someone else mentioned a rat. I know the very thought of these can make people shudder, but they do make excellent pets. They can respond to their names,are handled easily and rarely bite. If kept in a big enough cage, they can also be litter trained. Are also v cheap to care for.
I agree that kids can only really benefit from having a pet. Would a dog be too much work? If you are allergic to them, poodles and bichon frise are fairly non-allergic.