Fedya’s name day is January 11, the feast of Theodosius the Cenobiarch.
Churches that follow the Old (Julian) Calendar keep the same feast on January 24 of the civil calendar. See everything commemorated on January 11.
The saint behind the name

Theodosius the Cenobiarch
January 11, Venerable, Θεοδόσιος
St. Theodosius the Cenobiarch was born in Cappadocia in the fifth century and came as a young man to the Holy Land, where he embraced the ascetic life. Near Bethlehem he founded a great monastery of the common life, and he was appointed head of all the cenobitic monks of Palestine, whence his title Cenobiarch.
Name days: Theodosios, Theodosius, Fedya, Ted
Also celebrating on January 11
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.