Does performing Hajj on behalf of the questioner's father absolve the father of his obligatory Hajj, and which is more befitting: Hajj on behalf of the father, or Hajj on behalf of the maternal grandmother out of obedience to her, and is it permissible for the questioner to lie to his mother with the intention of performing Hajj on behalf of his father?

If your father had Hajj performed on his behalf, or if Hajj was not obligatory for him in the first place, then it is preferable to perform Hajj on behalf of your grandmother to please your mother and to fulfill the obligation for her, if Hajj was indeed obligatory for her. You will not be sinful if you insist on performing Hajj on behalf of your father, but it is not permissible to perform Hajj on behalf of your father and then tell your mother that you are performing Hajj on behalf of your grandmother.