Eugenia’s name day is December 24, the feast of Eugenia of Rome.
Churches that follow the Old (Julian) Calendar keep the same feast on January 6 of the civil calendar. See everything commemorated on December 24.
The saint behind the name

Eugenia of Rome
December 24, Martyr, Ευγενία
The holy virgin martyr Eugenia was the daughter of a distinguished Roman family in the third century. Coming to faith in Christ, according to tradition she fled a life of privilege, entered monastic life disguised as a man, and became known for holiness and healing; after many trials, in which her identity and her innocence were revealed, she returned to Rome and was put to death for the faith in the persecutions of that age.
Also celebrating on December 24
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.