I have a friend who had sex outside of marriage and she had a child out of it. She feels very strongly about not wanting the father to be in the child's life at all because he's not paying child support, he's not coming to visit his son as the court advised, and because of the terrible things he has done to her and her family. We were discussing that if she becomes involved with another man and they got married, would it be right in God's eyes for the husband against his wife wishes to tell her son that he's not his actual father, or let the child believe he's the father. Also, is it right for her to not want or allow the father to be in the child's life at all? Could you give me scriptural advice on this matter?
Just because a man and woman conceive a child, it doesn't not follow that they will make good parents for the child. Biology has nothing to do with being a father or a mother, other than that most become fathers and mothers through biology.
You make it sound as if it is this woman's fault that the man she sinned with does not want to be involved in his own child's life. If the man who was involved in the fornication is not acting as a father should, forcing the issue will not make him more responsible or a good dad for the child. He made his choice and unless he repents of his sins, it is likely best that he isn't involved in the child's life.
A small child doesn't need to know that one or both of his parents did not contribute to his DNA. The child doesn't care. All a child wants is for someone to love him and care for him. When he is older, he might ask and that would the time to explain. What he needs to know are the facts: his mother made a big mistake in having sex outside of marriage; worse, she did so with a man who was not a good man; and that rather than continue the mistake she sought out a good man to marry and be her child's father.
It should not be forgotten that our relationship with God is described as one of adoption. "But when the fullness of the time had come, God sent forth His Son, born of a woman, born under the law, to redeem those who were under the law, that we might receive the adoption as sons. And because you are sons, God has sent forth the Spirit of His Son into your hearts, crying out, "Abba, Father!" Therefore you are no longer a slave but a son, and if a son, then an heir of God through Christ" (Galatians 4:4-7). It is a statement of extra special love that a parent chooses to adopt a child and make him as if he was his own.