First of all Salat is not a Prayer. There is difference between Prayer and Salat.
DUA (Prayer) means Ask/invoke/Supplicate to Allah for worldly benefits that may be material or social for oneself and others.
SALAT means to follow closely the divine laws of Allah in our daily lives by taking heed (Rukku) on the message from Allah and surrendering (Sujjud) to the commandments of Allah.
Example: 14:40 (Abraham said) “My Lord, let me establish (Muqeema) the SALAT (Al Salati), and also from my progeny. Our Lord, and accept my PRAYER (DuAAi).”
THE WORD TAHAJJAD CAME FROM THE ROOT WORD HA-JIMM-DA
ha-Jiim-Dal = to sleep watch, remain awake. hajjada – to awake from sleep, pray in the night. ahjad – to lay the neck upon the ground (camel). tohajjad – remain awake.
tahajjada vb. (5) impv. 17:79
THE WORD TAHAJJAD IN THE QURAN
17:78 Establish (AQIMI O MUHAMMAD) the Salat at the Sinking (Duluk) of the sun, until the darkening of the night (Salat Isha); and the Qur’an at dawn (Salat Fajir) the Qur’an at dawn is witnessed.
17:79 And from the night, you shall remain awake (fatahajjad) with it (bihi -The Quran) as an addition (Nafilatan) for yourself (Laka-O Muhammad), perhaps your Lord would grant you a station that is praiseworthy.
PROPHET MUHAMMAD WAS COMMANDED BY ALLAH TO ESTABLISH THE NIGHT SALAT SESSION 73:1-8 & 73:20
73:1 O you who are covered with your garments.
73:2 Stand the night except for a little.
73:3 Half of it, or a little less than that.
73:4 Or a little more, and arrange the Quran in its arrangement.
73:5 We will place upon you a saying which is heavy.
73:6 The time of the night is more effective and better for study.
73:7 For you have many duties during the day.
73:8 And remember the name of your Lord, and devote to Him completely.
SPECIAL NIGHT TIME SALAT SESSION FOR THE PROPHET AND THE GROUP OF BELIEVERS
73:20 Your Lord knows that you rise a little less than two thirds of the night, and half of it, and one third of it; and a group of those who are with you. And Allah measures the night and the day. He knows that you will not be able to keep-up, so He pardons you. SO READ WHAT IS MADE EASY OF THE QURAN. He knows that there will be sick among you, and others that go forth in the land seeking from the bounty of Allah, and others who are fighting in the cause of Allah, SO READ WHAT YOU CAN OF IT. And Establish (AQIMU) the Salat and contribute towards purification and lend Allah a loan of righteousness. And whatever you put forth yourselves, you will find it with Allah, for it is better and a greater reward. And seek the forgiveness of Allah, for Allah is Forgiving, Merciful.
Tahajjad was a special Salat Session for Prophet Muhammad and the believers at the time of Prophet Muhammad.
During this salat session Prophet Muhammad was told to arrange the Quran in its arrangement 73:4 and study the Quran.
Also Prophet Muhammad was training the believers who were becoming Hafiz (Memorizer) of the Quran at night as mentioned in verse 73:20.