My children receive social security from their father. We are no longer married. Now he wants to sign off all his rights to the children. Would this stop their money?

There are two answers -

First, he CANNOT "sign off all his rights to the children" unless someone else adopts them. If they are adopted, then he is no longer their legal father and does not have to support them.

Second, he can't stop the SS they collect through his benefits UNLESS someone adopts the children.