The wisdom behind the mention of “ten nights” instead of ten days
Question

This question came to the mind of one of my relatives. It is: what is the wisdom behind Allah, may He be exalted, saying “By the ten nights” [al-Fajr 89:2] – even though the virtues of the (first) ten days of Dhu’l-Hijjah are in the days, not in the nights? Undoubtedly Allah is Most Wise.
Praise be to Allah.Allah, may He be exalted, says (interpretation of the meaning):
“By the dawn; By the ten nights” [al-Fajr 89:1-2]. 
There is a difference of opinion among the scholars as to what is meant by the ten nights mentioned in this oath. 
~1~ The majority of scholars are of the view that it is the first ten days of Dhu’l-Hijjah. In fact, Ibn Jareer (may Allah have mercy on him) narrated that there was scholarly consensus to that effect. He said: This refers to the (first ten) nights of Dhu’l-Hijjah, because there was consensus among the scholars of tafseer to that effect. End quote from Tafeer Ibn Jareer (7/514) 
Ibn Katheer (4/535) said: What is meant by the ten nights is the (first) ten of Dhu’l-Hijjah, as was stated by Ibn ‘Abbaas, Ibn az-Zubayr, Mujaahid and others among the earlier and later generations.
Here there arises the question that you mentioned, which is: what is the wisdom behind mentioning the nights and not the days? This may be answered as follows: 
The days are referred to as “nights” because the Arabic language is flexible and may use the word nights to refer to days, or the word days to refer to nights. The Sahaabah and Taabi‘een usually used the word nights to refer to days, to such an extent that they used to say “We fasted five”, referring to nights, even though the fast occurs during the day. And Allah knows best. 
This was stated by a number of scholars, including Ibn al-‘Arabi in Ahkaam al-Qur’an (4/334) and Ibn Rajab in Lataa’if al-Ma‘aarif (470). 
~2~ Some of the scholars are of the view – which was also narrated from Ibn ‘Abbaas (may Allah be pleased with him) – that what is meant by the ten nights is the last ten nights of Ramadan. They said: Because the last ten nights of Ramadan include Laylat al-Qadr, of which Allah (st) says (interpretation of the meaning):
“The night of Al-Qadr (Decree) is better than a thousand months” [al-Qadr 97:3]
“We sent it (this Quran) down on a blessed night ((i.e. night of Qadr, Soorah No: 97) in the month of Ramadan, the 9th month of the Islamic calendar). Verily, We are ever warning (mankind that Our Torment will reach those who disbelieve in Our Oneness of Lordship and in Our Oneness of worship)
Therein (that night) is decreed every matter of ordainments” [ad-Dukhaan 44:3, 4]. 
This view was favoured by Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him), because it is in accordance with the apparent meaning of the verse. 
See: Tafseer Juz’ ‘Amma by Shaykh Ibn ‘Uthaymeen.