Technically, we’ll start with before Mass, the recommendation to pray the Office of Readings:
110. Where possible, the Church desires that the faithful should prepare for the celebration of Midnight Mass on the 24 December with the Office of Readings(Cf. GILH 215). Where such is not possible, it may be opportune to arrange a vigil of hymns, readings, and elements drawn from popular piety.
This was a surprise when I read the document, I admit. Sunday, week one (where major feasts default) assigns Psalms 1, 2, and 3 in Christian Prayer. Psalm 2 fits, certainly. I would think one of the kingly Psalms also (93-99) over even the 145th.
Hymns and readings drawn from the Scriptures are more common. I like the Christmas homily of St Isaac the Syrian as a source from the writing of the saints.
111. At Midnight Mass, an event of major liturgical significance and of strong resonance in popular piety, the following could be given prominence:
• at the beginning of Mass, the proclamation of the Savior’s birth according the formula contained in the Roman Martyrology could be made in song;
• the prayer of the faithful should really be universal, and where appropriate, use several languages; and the poor should always be remembered in the presentation of the gifts;
• at the end of Mass, the faithful could be invited to kiss the image of the Child Jesus, which is then placed in a crib erected in the church or somewhere nearby.
These suggestions are striking. Many parishes sing the Christmas Proclamation–it has been promoted at least since the 80’s by many progressive liturgy outlets.
The suggestion to use several languages in the prayer of the faithful made me sit up and take notice. Why do you think this is especially good for Midnight Mass?
A collection for the poor, yes.
Veneration of the figure of the infant Jesus. I have never seen that happen before. What would the modern version be, selfies with the Baby?
The Directory on Popular Piety and the Liturgy is online at the Vatican site.
Catholic Sensibility 
The procession of the Santo Bambino to the presepe is a very prominent Italian practice at Christmas Midnight Masses. The Pope (if in good shape) does this at St Peter’s, IIRC.
I find it interesting how people forget the context that originally made Midnight Mass popular: the Eucharistic fast, which ran from midnight until Pius XII shortened it to 3 hours in the 1950s. The midnight fast meant that people could indulge in a meatless Christmas Eve feast and go right to Midnight Mass and take communion. (Christmas Eve was, like all major vigils, a penitential day of fast and abstinence as a formal matter until 1959, when it was formally allowed to be anticipated on December 23 – however, there had been some indults with that effect before that time, and IIRC this was also previously the case when Christmas Eve itself fell on a Sunday, but I could be wrong about that.)
I’ve seen that reference to the Office of the Reading in the Ordo, but didn’t know its origin. I see it is from the GILH. I’ve only done it once publicly: people preferred Lessons and Carols.