Efthimia’s name day is January 20, the feast of Euthymios the Great.
Churches that follow the Old (Julian) Calendar keep the same feast on February 2 of the civil calendar. See everything commemorated on January 20.
The saint behind the name

Euthymios the Great
January 20, Venerable, Ευθύμιος
St. Euthymios the Great was born in Melitene of Armenia in the fourth century and became one of the great fathers of the Palestinian desert. Settling in the wilderness east of Jerusalem, he founded a famous lavra, guided multitudes of monks, and formed disciples who themselves became pillars of monasticism, among them St. Savas the Sanctified.
Name days: Efthimios, Efthimis, Euthymios, Efthymios, Efthymis, Thymios, Efthymia, Efthimia, Thymia, Effie
Also celebrating on January 20
These names share the feast, many of them other forms of the same name:
Efthimios, Efthimis, Euthymios, Efthymios, Efthymis, Thymios, Efthymia, Thymia, Effie
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.