What must a young man do about his son born of adultery, given that the mother refuses to hand him over and demands money, and he is still a student financially dependent on his family?

Your friend committed a terrible sin, and we ask Allah to accept his repentance. As for his child born out of wedlock (zina), legally, he is not attributed to him, and he is not obligated to provide for his expenses or housing. Rather, the child is attributed to his mother, and she is responsible for his expenses, or the public treasury (Bayt al-Mal) if she is unable. There is no harm in him giving her whatever he wishes for this child, as an act of kindness and a gift, not as an obligation. He does not have the right to take the child from his mother without her consent.