"The best prayer after the obligatory prayers is the one during the depths of the night." [Muslim]
"You should perform the late-night prayer for verily it was the custom of the pious people before you. Certainly, the late-night prayer is an act that takes one closer to Allah, keeps one away from sins, expiates for evil deeds and repels disease from the body." [Tirmidhi]
There are many aspects that make the late-night prayer special. First and foremost, it is a prayer. The best action or matter is prayer.
Second, al-Haitami points out that, in general, the prayer performed in the late-night is more virtuous than the voluntary prayers during the day because it is further from ostentation and being done for show. In other words, the intention behind the late-night prayers should be a pure one. Furthermore, it is easy to have the fear of Allah and concentrate on the prayer in the late-night prayer as there are fewer disturbances at the time.
Thirdly, it makes the person realize his real purpose and goal in this life. He is getting up in the middle of the night and forsaking his sleep because he realizes that his goal is not the pleasure or relaxation of this world. Instead, he has a much greater purpose and goal in life. For that reason, he is getting up to pray when the rest of mankind are either sleeping or committing sins under the cover of night.