[A father’s bequest to his son that he perform Hajj on his behalf]

Answer: If the son is able [financially and physically] to perform Hajj for himself, then it is not appropriate for him to perform Hajj on behalf of his father or on behalf of anyone else. He must first perform Hajj for himself, then [after that] perform Hajj for his father or for someone other than his father.
If the son knows from his father’s state that he chose him for Hajj only because of his righteousness, and that his only motive in appointing him was that righteousness, then there is no objection to the son hiring one of the righteous, sound men [to perform Hajj on his father’s behalf].
But if the father appointed his son for some purpose other than mere righteousness, in addition to that righteousness, then it is not permissible for the son to hire another man for Hajj. The obligation upon the son is that he first perform Hajj for himself, and then perform Hajj for his father. This is what is indicated by the words of the scholars of the madhhab.
Yes, it may be permissible for the son to hire a hired person to perform Hajj on behalf of his father in a case where the son thinks it is unlikely that he will be able to perform Hajj for his father later on, due to illness, or for financial reasons, or for political reasons, or the like. In such a circumstance for the son, it is valid for him to hire someone to perform Hajj on behalf of his father, in light of His, the Exalted’s, saying: “Allah does not charge a soul except [with that within] its capacity” [al-Baqarah:286], and His, the Exalted’s, saying: “So fear Allah as much as you are able” [al-Taghābun:16], and the hadith of the Prophet (may Allah bless him and his family and grant them peace): “When I command you with a matter, do of it what you are able.”

Source: Min Thimār al-ʿIlm wa al-Ḥikmah vol.2