Optional night prayer is a confirmed Sunnah and it is the best prayer after the obligatory ones.
The Prophet (may Allah exalt his mention) said:
The best prayer after the obligatory prayer is the optional night prayer.
The best time would be the last third of the night as came in Sahih Muslim:
Abu Huraira reported Allah's Messenger (ﷺ) as saying: When half of the night or two-third of it is over. Allah, the Blessed and the Exalted, descends to the lowest heaven and says: Is there any beggar, so that he be given? Is there any supplicator so that he be answered? Is there any beggar of forgiveness so that he be forgiven? (And Allah continues it saying) till it is daybreak.
How does allah descend ? Only allah knows. However, Allah descends in a way that suits his majesty and greatness.
To find the last third of the night: Calculate how many hours between Maghrib and Fajir. e.g: 9 hours. So the night is 9 hours. Now divide it by 3. So the last third is the 3 hourse before Fajir Athan