Onyfrios’s name day is June 12, the feast of Onuphrios the Great.
Churches that follow the Old (Julian) Calendar keep the same feast on June 25 of the civil calendar. See everything commemorated on June 12.
The saint behind the name

Onuphrios the Great
June 12, Venerable, Ονούφριος
St. Onuphrios the Great was one of the greatest hermits of the Egyptian desert, living in the fourth century. Leaving his monastery for the inner wilderness, he dwelt alone for some sixty years in utter poverty, unseen by men, clothed only in his own hair and a girdle of leaves, sustained by God in prayer.
Name days: Onyfrios
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.