Answer
Wa `alaykum as-salam wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- Even though you are allowed to perform Hajj on behalf of any person other than your parents, there might be no need to perform Hajj on behalf of the Prophet (peace and blessings be upon him).
2- None of the early good predecessors was reported to have performed Hajj on behalf of the Prophet (peace and blessings be upon him), and we are to follow the example of good righteous Muslims.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Even though we are allowed to perform Hajj on behalf of people other than our parents and close relatives, there is no need for us to perform Hajj on behalf of the Prophet (peace and blessings be upon him).
This is not something that is recommended to do, for there is no known precedent for it in the perfect examples of the pious generations whose examples we are to emulate as much as possible.
You can, however, perform Hajj on behalf of any other Muslim, male or female, who may have never performed Hajj, or you can simply do it as an extra act of devotion on behalf of yourself.
It is even better if you donate the money for far more important causes such as helping a poor student, or to help refugees.
Allah Almighty knows best.
Editor’s note: This fatwa is from Ask the Scholar’s archive and was originally published at an earlier date.