Melanie’s name day is December 31, the feast of Melania the Elder.
Churches that follow the Old (Julian) Calendar keep the same feast on January 13 of the civil calendar. See everything commemorated on December 31.
The saint behind the name

Melania the Elder
December 31, Holy Woman, Μελανία
St. Melania was a Roman noblewoman of one of the wealthiest families of the empire who, in the era of the great flowering of monasticism, renounced rank and riches for the sake of Christ. She gave her vast possessions to the poor and to the churches, freed those enslaved to her household, and devoted herself to asceticism, prayer, and the service of the monks and holy places of the East, ending her days in the Holy Land.
Also celebrating on December 31
These names share the feast, many of them other forms of the same name:
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.