Reference: Majmoo’ Fataawa wa Rasaa.il ‘Abdul-‘Azeez ibn Baaz – Volume 16, Page 404
Question: If a man died without having instructed anyone to perform hajj on his behalf [after his death], then is he relinquished of this obligation if his son performs hajj on his behalf?
Response: If his Muslim son – who has already performed hajj on behalf of himself – [now] performs hajj on his [father’s] behalf, then by that, he (the father) is relinquished of his obligation.
And similarly, if someone other than his son – from amongst the Muslims who has already performed hajj on his own behalf – were to perform hajj on behalf of him (the father).
[This is] based upon what has been authentically reported from Ibn ‘Abbaas (radhi-yAllaahu ‘anhumaa) that a woman said:
“O Messenger of Allaah! The obligation of Hajj enjoined by Allaah on His worshipers has become due on my father and he is old and unable to perform hajj, and he cannot sit firm on the mount; should I perform Hajj on his behalf?” He replied:
«Yes, perform hajj on his behalf.» – transmitted by at-Tirmidhee no.928
And there are other ahaadeeth on this topic that indicate what we have mentioned [above].