Traditionally, 3 pm in the afternoon is the holy hour of mercy when Jesus died, and 3 am in the morning is supposed to be its opposite - the time when Satan is taking the most souls to hell. Some Catholics like to pray at 3 am because they believe that is the most important time to help souls and oppose Satan. However, this is a private devotional practice, and while it is permitted by the Church, it is not something officially promoted by the Church. Priests would generally tell you that a prayer is good no matter what time of day it is said, especially since God is timeless and does not follow earthly time.
Those Catholics who pray the Liturgy of the Hours (primarily clergy and religious orders, but also some lay people) may have, or had in the past, some regularly scheduled prayer at 3 am. It’s my understanding that in recent revisions, the set times are now more broad ranges so the prayer may not be said precisely at 3 am but within some general time range around that time. As these people are saying prayers all throughout the day, the one at 3 am is not really more special or different than the ones said at other times.
