'Dermatitis' is a very general term. While it can refer to allergy-related symptoms ('allergic contact dermatitis') it's far more likely that a child will have a form of eczema (e.g. infantile atopic eczema). Such conditions are not affected by contact with pet hairs, so you're not restricted in your choice of pets.
Speak to your doctor first (to confirm that there really is no problem) and then (as long as your family lifestyle permits it) get a dog. While I love cats, youngsters find them extremely frustrating because they don't openly return the love that you show them. Dogs, on the other hand, love their owners even more than their owners love them and kids naturally bond with them.
Chris