Is there a supplication for memorizing the Quran, and what is the (proper) methodology to memorize the Qur’an?
I don’t know of any supplication for memorizing the Noble Qur’an; except for one Hadeeth which has been narrated on the Prophet (صلى الله عليه و سلم) in which he (supposedly) taught ‘Ali (رضى الله عنه) (a supplication for the memorization of the Qur’an). There is doubt or question with regards to the authenticity of this Hadteeh .
Ibn Katheer (رحمه الله) said about this Hadeeth,
The strangeness of this Hadeeth is abundantly clear.
Shaikh As Sayid Muhammad Rasheed Ridaa (رحمه الله) said commenting upon this,
Rather, the style and nature of language (of this Hadeeth) is that of the fabricated (narrations), it is not the style or nature of the language of the most eloquent of mankind, Muhammad (صلى الله عليه و سلم), ‘Ali (رضى الله عنه) or anyone of their time.
Imaam Ath Thahabee (رحمه الله) said:
This Hadeeth is weak and contradictory going against that which is more authentic.
However, the way in which an individual will memorize the Qur’an is by being consistent (upon memorizing it). People (generally) have two ways of going about memorizing (the Qur’an).
The First: is to memorize verse by verse, or two or three verses at a time, depending on the length of the verses.
The Second: is to memorize page by page.
People differ with regards to the ways in which they memorize. Some of them prefer to memorize page by page, repeating it until it is memorized. While others prefer verse by verse, repeating them until they are memorized. Once memorized, they will then move on to the next verse, and so on.
Furthermore, it is incumbent whether, (one memorizes) by the first or second method, that they do not proceed (to the next verse or page) until they have excellently memorized (that verse or page that they are currently on), in order that they do not build upon a weak foundation. It is incumbent that they go over what they memorized daily, in particular in the morning time (after Fajr). If they know that they memorized well (what they are currently on), then (at that point) they should proceed to a new lesson.
 This Hadeeth has been collected by At-Tirmidhee in his Sunnan, in the chapter entitled “Supplications” (under the) subchapter (title) “Supplication for Memorization” #3081. Al ‘Allaamah Imaam Al Albaanee has graded this Hadeeth unauthentic (da’eef) in Silsilatud Da’eefah #3374, as well as in Da’eef At-Tirmidhee.