She was married when the kid was born but the kid was not the husbands. The kid was given the fathers last name but now the mom wants to change the kids last name to hers. How do she go about doing this?

She informs the legal father (her ex husband) and if he agrees then she goes to court with his blessings to change it. If not then she will have to fight for the name change and depending on the age of the child the courts may be highly reluctant to approve of a change.