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The girlfriend has no standing to bring a custody action regarding the child, because she is not related to the child in any way. Only your son can bring a custody action, and if he was never married to the mother and they did not sign an acknowledgment of paternity at birth, he is not presumed to be the father. He must first establish paternity through genetic testing. Only that can he request custody rights.
It is not likely that he will gain sole custody unless there are serious problems with the mother that would put the child in jeopardy. He may be able to obtain visitation rights. The court will look to what is in the best interest of the child. If the court determines, as it often does, that it is in the best interest of the child to have a relationship with the father, it may order visitation rights for him. If the mother is concerned for the child's safety due to his mental disability, she may request that visitation is supervised.