My mother made a promise to Allah that if she gets a daughter she will make one of her sons a Qur'an memorization, i.e. One of her sons will memorize the Qur'an. She got twins a son and daughter. She tried to put one of my brothers for Qur'an memorization one year. But it didn't turn out well and he could not do it. My question is "Is there anyway to fulfill this promise other than doing memorization". Can my grand children do memorization instead of my children?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions. If your mother has vowed to do her best and has fulfilled her promise by doing everything she can to make her child recite the Qur'an but in vain, then she is not supposed to expiate for this as she has done what she could. Expiation would have been obligatory on her if she had been short of using all possible means to fulfill her vow (to make one of her male children memorize the Qur'an). If she had vowed to achieve this, then this is something beyond her ability the Prophet (Sallallahu Alaihi wa Sallam) said: "And there is no vow on the child of Adam in anything that he does not own" [Reported by Imams al-Bukhari and Muslim ], then there is nothing on her. Allah knows best.