You certainly can change names, whether your own or your children. Procedure is to see a lawyer to do a "deed poll" ie a legal doc drawn up by the lawyers to state and announce name change and what the the new name is. Such preodure are usually quite fats, take probably around 2 weeks and does not cost alot. Probably a few hundred dollars one-time fee.
I am not sure about legal rights over kid if unmarried though..