Praise be to Allah.
It is better for him to do Hajj on his own behalf, because this is the basic principle, and he can make du’aa’ for himself and for others among his relatives and all the Muslims. But if one or both of his parents did not do the obligatory Hajj, then he may do Hajj on their behalf after doing Hajj for himself, as a way of honouring them and treating them kindly when they are incapacitated or after they have died. But he should do Hajj or ‘Umrah for each of them separately; he cannot combine them both in one ‘Umrah or one Hajj. End quote.
Fataawa al-Lajnah al-Daa’imah, 11/66
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: A woman wanted to do Hajj on behalf of her mother who had died, but her mother had done the obligatory Hajj. Which is better: to do Hajj on her behalf or to make du’aa’ for her?
It is better for her to do Hajj for herself, and to make du’aa’ for her mother, because the Prophet (peace and blessings of Allaah be upon him) said: “When a man dies, all his good deeds come to an end except three: ongoing charity (sadaqah jaariyah), beneficial knowledge, or a righteous son who will make du’aa’ for him.” He did not say: a righteous son who will do Hajj on his behalf or fast on his behalf or give charity on his behalf or pray on his behalf. If someone were to ask us: which is better, to pray and give the reward to my father, or to give charity and give the reward to my father, or to make du’aa’ for my father? We would say: it is better to make du’aa’ for your father, because the Messenger (peace and blessings of Allaah be upon him) knew more than us, and was more sincere than us, and more eloquent than us, and he did not say, a righteous son who will do good deeds on his behalf; rather he said: a righteous son who will make du’aa’ for him. This is what the Prophet (peace and blessings of Allaah be upon him) taught us. End quote.
Fataawa Ibn ‘Uthaymeen, 21/251.