Praise be to Allah.
I do not know of any du’aa’ for memorizing Qur’aan, except one hadeeth. It was narrated that the Prophet (peace and blessings of Allaah be upon him) taught it to ‘Ali ibn Abi Taalib (may Allaah be pleased with him), but there is some dispute as to how sound it is. Ibn Katheer said concerning it: it is ghareeb, rather munkar (i.e., it is not saheeh).
Al-Sayyid Muhammad Ridaa said, commenting on it: its style is that of the fabricated reports, not the style of the most eloquent of mankind, Muhammad (peace and blessings of Allaah be upon him) or the style of ‘Ali ibn Abi Taalib (may Allaah be pleased with him), nor the style of their age.
Al-Dhahabi said this is a shaadhdh (odd) munkar hadeeth.