As-salaamu `alaykum wa rahmatullaah
Memorisation of the Qur’aan guides you to the Straight Path
We know for sure that if a youth keeps himself busy in the memorisation of the Qur’aan from a young age, then that will keep him busy from falling into sin and disobedience. From there, he will be going from memorisation to revision and from studying the tafseer (interpretation) of those verses to striving upon learning the Asbaab an-Nuzool (causes for revelation). Like that he’ll find himself moving from knowledge to knowledge until at the end of the matter, he becomes a scholar from the righteous scholars of this Ummah
Memorising Qur’aan means taking advantage of the moments of life
Indeed, the Prophet (sallallaahu ‘alayhi wa sallam) ordered us to advantage of every moment of our lives and use it in the obedience of Allaah. He (sallallaahu `alayhi wa sallam) said,
“Take benefit of five before five:
your youth before your old age,
your health before your sickness,
your wealth before your poverty,
your free-time before your preoccupation,
and your life before your death.” 
In a time when the people of this world are busy with their Dunya and its vanities, and they make their gatherings one of amusement, heedlessness and slander – the people of Qur’aan are taking advantage of every moment and living by the Qur’aan, in heart and soul, reciting it, memorising it, pondering over it and acting upon it.
Beloved brothers and sisters, indeed the nafs (soul) commands evil and if you do not busy it in obedience, it will busy you in disobedience. Time is like a sword, if you do not cut it, it will cut you – so take advantage of life… memorise the Book of Allaah and act upon it.
The Qur’aan is a cure for the hardening of the heart
Many people complain about the hardness of their hearts and they ask for the cure to it, so we say to them: Indeed, the greatest way to cure the hardness of the heart is to draw closer to the Book of Allaah ‘azza wa jall.
“Allah has sent down the best statement, a Book, its parts resembling each other in goodness and truth, oft-repeated.
The skins of those who fear their Lord shiver from it.
Then their skin and their heart soften to the remembrance of Allah.
That is the guidance of Allah.
He Guides therewith whom He pleases and whomever Allah sends astray, for him there is no guide.” [Az-Zumar: 23]
The Qur’aan increases your Imaan
Whoever wishes for an increase in Imaan day after day, then upon him is (recourse) to the Book of Allaah, for He (subhaanahu wa ta’aala) has said,
“The believers are only those who, when Allah is mentioned, feel a fear in their hearts and when His Verses are recited unto them, they increase their Faith; and they put their trust in their Lord (Alone)” [Al-Anfaal: 2]
He also said,
“And whenever there comes down a Surah (chapter from the Quran), some of them (hypocrites) say: “Which of you has had his Faith increased by it?” As for those who believe, it has increased their Faith, and they rejoice.” [At-Tawbah: 124]
For whenever you read in the Qur’aan how Allaah gave victory to His slaves, the Prophets and Messengers (`alayhumusalaam) and brought defeat to the disbelievers, you’ll increase in Imaan. When you read about the Promise of Allaah to His believing servants, the promise of Paradise and pleasure… and His stern warnings of anger and hellfire to the disbelievers, you’ll increase in Imaan.
Narrated from Jundub (radhi’Allaahu `anhu) that he said, “We were young youth with the Messenger of Allaah (sallallaahu `alayhi wa sallam), so we learnt Imaan before we learnt Qur’aan, then we learnt Qur’aan and it increased our Imaan.” 
Indeed in the Messenger of Allaah you have a great example
Allaah Ta’aala said,
“Indeed in the Messenger of Allah (Muhammad SAW) you have a good example to follow for him who hopes in (the Meeting with) Allah and the Last Day and remembers Allah much.” [Al-Ahzaab: 21]
The Prophet (sallallaahu `alayhi wa sallam) used to memorise the Qur’aan and he used to be continuous in its recitation. Jibreel (`alayhisalaam) used to review it with him once every year and in the year that he (sallallaahu `alayhi wa sallam) was taken (i.e. passed away), he came to him twice. So whoever memorised the Qur’aan, then he has indeed taken the beloved as an example.
Memorisation of Qur’aan is from the special characteristics of this Ummah
Hifdh al-Qur’aan is from the special characteristics of the Ummah of Muhammad (sallallaahu `alayhi wa sallam). Ibn al-Jazari (rahimahullaah) said: “The transmission of Qur’aan by reliance upon memorisation (of by heart) and not upon script in the mus-haf and in books – this is from the most noble of characteristics given to this Ummah by Allaah Ta’aala.” He also said, “Allaah ta’aala has informed us that the Qur’aan does not need to be confined – in preserving it – to a page that can be washed with water. Instead it is read in all states as has been narrated in the description of this Ummah “Their scriptures are in their hearts” – and this is different to the people of the Book who do not memorise it except in (referring to) the Books, they do not read it except by looking at it and it is never off by heart.
 al-Haakim and al-Bayhaqi – saheeh al-Jaami’ (1077)
 Nuzhat al-Fudhalaa (383/1)