He vowed that he would not get married until his mother memorized a portion of Qur’aan, but it is very difficult for her
It is not permissible for anyone to oblige his mother to do something that is not obligatory in sharee’ah and causes difficulty for her. So long as your mother is not able to memorize except with great difficulty, then you have no right to oblige her to do that. You have to apologize to her and try to please her, and offer kafaarat yameen (expiation for breaking an oath), because the vow that is aimed at encouraging a person to do a specific action comes under the same ruling as an oath (yameen). If you offer the expiation for breaking an oath, then the vow is no longer binding, and your mother does not have to memorize anything that she cannot.
Ibn Qudaamah (may Allaah have mercy on him) said, discussing different types of vows: The vow that is made in the form of an oath to encourage a person to do something or to stop them doing something, when the intention is not to make a vow or do an act of worship, comes under the same ruling as an oath (yameen). End quote from al-Mughni (10/67).
You have to please your mother and offer the expiation for breaking an oath, and get married.
For detail on the expiation for breaking a vow, please see the answer to question no. 45676.
And Allaah knows best.