Can a Woman Perform Hajj on Behalf of Her Father?

Yes, it is permissible for a woman to do Hajj on behalf of a man. 

Al-Bukhari (1513) and Muslim (1334) narrated that ‘Abd-Allah ibn ‘Abbas (may Allah be pleased with him) said: A woman from Khath-am came and said: “Allah’s command to the people to perform Hajj has come when my father is an old man and cannot sit firmly in the saddle. Can I perform Hajj on his behalf?” He (the Prophet (peace and blessings of Allah be upon him)) said: “Yes.” That was during the Farewell Pilgrimage.

Ibn Hazm said in al-Muhalla (5/317): 

“It is permissible for a woman to perform Hajj on behalf of a man or a woman , and for a man to perform Hajj on behalf of a woman or a man, because the Prophet (peace and blessings of Allah be upon him) commanded the Khath-ami woman to perform Hajj on behalf of her father, and he commanded a man to perform Hajj on behalf of his mother, and another man to perform Hajj on behalf of his father. There is no text which forbids any of that. Allah says (interpretation of the meaning): “and do good” [al-Hajj 22:77], and this is good. It is permissible for everyone to do it on behalf of everyone.”

Ibn Qudamah said in al-Mughni (5/27): 

“It is permissible for a man to be appointed to do Hajj on behalf of a man or a woman , and a woman on behalf of a man or a woman, according to the majority of scholars. We do not know of any difference of opinion, except al-Hasan ibn Salih, who regarded it as makruh for a woman to do Hajj on behalf of a man. Ibn al-Mundhir said: this is a failure to recognize the apparent meaning of the Sunnah, because the Prophet (peace and blessings of Allah be upon him) commanded a woman to do Hajj on behalf of her father.”

Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) was asked: is it permissible for a woman to do Hajj on behalf of her father even if she has brothers who are grown up? 

He replied: 

“It is permissible for a woman to do Hajj on behalf of her father , even if she has brothers who are grown up. Both men and women can act as deputies on behalf of others . Hence the woman from Khath-am asked the Prophet (peace and blessings of Allah be upon him)… And he quoted the hadith mentioned above, then he said: He (the Prophet (peace and blessings of Allah be upon him)) gave her permission to do Hajj, and she was a woman acting on behalf of a man. But it is essential for her to have a mahram on every journey, whether that is for Hajj or for any other purpose, and whether the woman is travelling to do Hajj for herself or on behalf of someone else.” (Fatawa Ibn ‘Uthaymin, 21/247)

And Allah knows best.