Is it permissible to accept money from someone in exchange for performing Hajj on behalf of their relative, knowing that the Ministry has fully covered the expenses of Hajj?

All praise is due to Allah.

Performing Hajj on behalf of someone else, using money, takes two forms: deputization (niyabah) and hiring (ijarah).

As for the deputy (na'ib): he does not take a wage, but rather takes the expenses for travel and Hajj, and he must return any surplus money unless the one who deputized him gives permission otherwise. If he performs Hajj for free, he must return all the money.

As for hiring (ijarah): in this case, the hired person takes the agreed-upon wage, even if he performs Hajj for free.

Scholars have differed regarding the permissibility of hiring for Hajj. Some have prohibited it, some have deemed it disliked, and some have permitted it, such as the Shafi'is.

The difference between deputization and hiring is that the deputy takes only the amount of his expenses, whereas the hired person agrees upon a specific amount that is not tied to his expenses.

Taking a wage for Hajj is not among the deeds of the righteous. So, whoever performs Hajj for free should perform Hajj for himself and apologize to his benefactor, or perform Hajj for his relative for free.