All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
You mentioned that the old son of your mother-in-law performed Hajj on her behalf last year. Thus, he dropped the obligation from her and freed her liability. Your wife wants to perform Hajj another time on her mother’s behalf. Now, you are asking about the ruling on that and whether the Hajj will be rewarded.
Actually, scholars had different opinions as regards the voluntary Hajj, whether or not it will be useful to the dead person. The preponderant opinion as indicated by a group of scholars is that the dead person benefits from the voluntary Hajj and its reward reaches him. We in Islamweb hold the same view.
It is not right that the daughter is not allowed to perform Hajj on behalf of her mother, since there is no difference between males and females in this regard. Hence, your wife can perform voluntary Hajj on behalf of her mother and the reward for Hajj will reach her mother, Allaah Willing. She is required to offer a Hady (Hajj sacrificial animal) if she performs Tamattu’ or Qiraan Hajj. However, if she assumes Ihraam for Ifraad Hajj, then she will not be required to offer a Hajj sacrificial animal. In this concern, no difference exists between a Hajj that a person performs for himself and a Hajj that he makes on behalf of someone else.
Allaah Knows best.