I think there may be a oral tradition of that the birth occurred at midnight. I cite this prayer for your consideration. Maybe someone else can confirm this oral tradition. So if Christ was born at midnight, I think the mass at midnight is very fitting. Traditions are important if it draws us to reflect on the reason for the tradition. Good question. I would like the reason for hte midnight mass and why we have mostly lost that tradition.
St. Andrew Christmas Novena Prayer
Hail and blessed be the hour and moment in which the Son of God was born of the most pure Virgin Mary, at midnight, in Bethlehem, in piercing cold.
In that hour, vouchsafe, O my God! to hear my prayer and grant my desires, through the merits of Our Saviour Jesus Christ, and of His Blessed Mother. Amen.
(It is piously believed that whoever recites the above prayer fifteen times a day from the feast of St. Andrew (30th November) until Christmas will obtain what is asked.)
Imprimatur
+MICHAEL AUGUSTINE, Archbishop of New York
New York, February 6, 1897