How can a mother with a green card get her son to usa?

Not necessary since the child is a US citizen by birth. Before the child is 18 years of age you need to apply for US citizenship certificate using Immigration Form N-600.

If the question is referring to the PARENT of the US-born child getting a green card, then there is no special way to do this: you will have to wait until your child is 21, after which they can then sponsor you for a green card. Merely being the parent of a US Citizen does not automatically entitle one to a green card immediately (or automatically).