My son lives with me and visit his mother a couple days out of the week. She decides she wants to have me up for child support and I was wondering could she do that?
Divorce / Separation Lawyer
You cannot "sign over your rights". Your child can be adopted by someone else which would terminate your parental rights and would eliminate the child support obligation.
If you have physical custody of the child, you would not be paying child support in the usual case. In shared custody cases, you still may owe child support. It sounds like you probably have no custody order though, and the other parent may dispute your claim about the amount of time that the child spends with you. For that reason, you should obtain a custody order ASAP to clarify that issue.