What is the jurisprudential ruling, and what is required of the husband in the event of a dispute between him and his wife, the cause of which is the father's insistence that his son remain close to him to serve him and lead him in prayer, which the wife considers a restriction on her life and demands that he move to another house?

Your brother is not obligated to obey his wife in moving from his father's house as long as the dwelling is independent and suitable, and she can stay with him. As for the father's request to his son to stay with him for some time to pray in congregation, the son must obey his father in this, and this delay does not harm the wife. Your brother should explain to his wife his father's right over him. If the wife insists on moving and this does not prevent caring for the father, then there is no objection to moving. However, if moving prevents him from honoring and caring for his father, then he should not obey her in that, and he must act wisely and not concede to her if he perceives that she is trying to distance him from his father, nor should he be affected by her threats.