Anysia’s name day is December 30, the feast of Anysia of Thessaloniki.
Churches that follow the Old (Julian) Calendar keep the same feast on January 12 of the civil calendar. See everything commemorated on December 30.
The saint behind the name

Anysia of Thessaloniki
December 30, Martyr, Ανυσία
The holy virgin martyr Anysia was a young Christian of Thessaloniki who lived at the turn of the fourth century, in the days of the persecution under Diocletian. Left wealthy by her parents, she gave her goods to the poor and lived in prayer and quietness. On her way to the gathering of the faithful she was stopped by a soldier who tried to drag her to a pagan sacrifice; when she confessed Christ and resisted, he ran her through with his sword.
Name days: Anysia
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.