Assalamualaikum ustaz. Is it permissible to offer the memorization of the Quran as one’s dowry?
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
According to the 4th Edition of Kamus Dewan, mahar is an obligated gift from the groom to the bride and it is named dowry. 
Mahar is also called al-sadaq and it is obligatory as soon as the solemnization is completed, regardless of whether the dowry is mentioned or not during the solemnization. If there is an agreement but it is denied or isn’t stated, the agreement is void and the dowry is still obligatory to be given. 
Syeikh Muhammad Al-Zuhayli said mahar according to jurists means a valuable thing that is obligatory to be given by the groom to the bride before marriage that makes the relationship between them permissible. The terminology mahar is also stated by al-Sodaaq in fiqh books and there are other names given for mahar and all of the terminologies are mentioned in al-Quran and hadith of the Prophet PBUH. 
Allah SWT states:
وَآتُوا النِّسَاءَ صَدُقَاتِهِنَّ نِحْلَةً
“And give the women [upon marriage] their [bridal] gifts graciously.” 
Ibn Kathir cited the opinion of Ibn Abbas, the wordالنحلة means mahar. While Ibn Zaid explained that it means something obligatory to be given to the wife due to marriage. A man is obligated to give the mahar to the woman as a gift willingly. 
The person who is commanded to give the dowry is the husband. Likewise, this is stated by Allah SWT:
فَمَا اسْتَمْتَعْتُم بِهِ مِنْهُنَّ فَآتُوهُنَّ أُجُورَهُنَّ فَرِيضَةً
“So, for whatever you enjoy [of marriage] from them, give them their due compensation as an obligation.” 
Next, Allah SWT states:
لَّا جُنَاحَ عَلَيْكُمْ إِن طَلَّقْتُمُ النِّسَاءَ مَا لَمْ تَمَسُّوهُنَّ أَوْ تَفْرِضُوا لَهُنَّ فَرِيضَةً
“There is no blame upon you if you divorce women, you have not touched nor specified for them an obligation.” 
Memorization of Quran as Dowry
Considering dowry, the name mahar, originally means an object. Except when he doesn’t own anything. For example, if a person has the expertise to teach the recitation of the Quran, then this is permissible.
However, originally, a valuable thing should be given. The reason is the Prophet PBUH said:
الْتَمِسْ وَلَوْ خَاتَمًا مِنْ حَدِيدٍ
“Try (to find something), even if it were an iron ring,” 
According to the above question, making the memorization of the Quran as mahar with the meaning of the husband teaching his wife al-Quran until she is able to recite it as stated in the hadith.
From Sahl bin Sa’ad RA, he said:
أَتَتِ النَّبِىَّ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ امْرَأَةٌ فَقَالَتْ إِنَّهَا قَدْ وَهَبَتْ نَفْسَهَا لِلَّهِ وَلِرَسُولِ اللَّهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ فَقَالَ: مَا لِى فِى النِّسَاءِ مِنْ حَاجَةٍ. فَقَالَ رَجُلٌ زَوِّجْنِيهَا. قَالَ: أَعْطِهَا ثَوْبًا. قَالَ لاَ أَجِدُ. قَالَ: أَعْطِهَا وَلَوْ خَاتَمًا مِنْ حَدِيدٍ. فَاعْتَلَّ لَه. فَقَالَ: مَا مَعَكَ مِنَ الْقُرْآنِ. قَالَ كَذَا وَكَذَا. قَالَ: فَقَدْ زَوَّجْتُكَهَا بِمَا مَعَكَ مِنَ الْقُرْآنِ
“A woman came to Allah’s Messenger (ﷺ) and said, “I present myself (to you) (for marriage). She stayed for a long while, then a man said, “If you are not in need of her then marry her to me.” The Prophet (ﷺ) said, “Have you got anything m order to pay her Mahr?” He said, “I have nothing with me except my Izar (waist sheet).” The Prophet (ﷺ) said, “If you give her your Izar, you will have no Izar to wear, (so go) and search for something. He said, “I could not find anything.” The Prophet (ﷺ) said, “Try (to find something), even if it were an iron ring. But, he was not able to find (even that) The Prophet (ﷺ) said (to him). “Do you memorize something of the Qur’an?” “Yes. ‘ He said, “such Sura and such Sura,” naming those Suras. The Prophet (ﷺ) said, “We have married her to you for what you know of the Qur’an (by heart).” 
In a Muslim narration, it is stated:
انْطَلِقْ فَقَدْ زَوَّجْتُكَهَا فَعَلِّمْهَا مِنْ الْقُرْآنِ
“Go, I have married her to you, and you teach her something of the Qur’an.” 
Imam al-Nawawi said: “In the hadith, there is evidence that a mahar can be in the form of teaching of the Quran.” 
In al-Mausu’ah al-Fiqhiyah al-Kuwaitiyah  it is stated:
“Majority of jurists from madhhab Maliki, Syafi’e and Hanbali stated that everything that is valuable or payment is permissible to be made as dowry, while madhhab Hanafi stated that dowry is any valuable property for people. If it is named as property, then the mahar is valid. If it is not a property, then the mahar is invalid.
The majority of scholars are of the opinion that it is of value and benefit.
Madhhab Syafi’e, Hanbali and Maliki according to the famous opinion stated that it is permissible to make anything of benefit as dowry according to their principles that anything that can be benefitted from: then it is valid to be made as mahar. It is valid for a husband to offer mahar such as a house, animal mount, or a slave for a year for the benefit of his wife, or husband offering his services as mahar to plant trees, or building a house or sew clothes, or offering his services for travel hajj and others for example.”
According to the above question, if the husband makes his memorization of the Quran as the mahar, then it is invalid for the memorization of the Quran of a man or husband is only beneficial for himself and not a benefit the husband can give to his wife as mahar. Except if the husband makes the mahar for him to teach the Quran to his wife, then this can be made into mahar according to the accurate opinion. The reason is teaching involves hardship, effort and time and usually, the process of teaching is valuable and people will pay for this service.
To close, we state that it is permissible to give mahar in the form of teaching his wife until she is able to recite the Quran. The reason is that teaching his wife the Quran is beneficial for the wife where she’ll eventually be able to recite the Quran properly and understands it. However, it should be reminded that the priority is by giving valuable things such as cash, gold, real estate and others. If he is incapable to do so, then the mahar can be in the form of teaching the Quran as stated in the hadith of Sahl bin Sa’ad RA that we have stated above.
May Allah SWT grant us understanding in practising all His commandments. We end with the supplication:
رَبَّنَا هَبْ لَنَا مِنْ أَزْوَاجِنَا وَذُرِّيَّاتِنَا قُرَّةَ أَعْيُنٍ وَاجْعَلْنَا لِلْمُتَّقِينَ إِمَامًا
“Our Lord, grant us from among our wives and offspring comfort to our eyes and make us an example for the righteous.”
 See Kamus Dewan Edisi ke-4, pg. 975.
 See al-Fiqh al-Manhaji, 4/75
 See al-Mu’tamad fi al-Fiqh al-Syafi’e, 4/109-110
 Surah al-Nisa’: 4
 See Tafsir al-Quran al-‘Azim, 2/235
 Surah al-Nisa’: 24
 Surah al-Baqarah: 236
 Narrated by al-Bukhari (5135)
 Narrated by al-Bukhari (4741)
 Narrated by Muslim (1425)
 See al-Minhaj Syarh Sahih Muslim, 9/214
 See al-Mausu’ah al-Fiqhiyah al-Kuwaitiyah, 39/155-156