Mitrofanis’s name day is June 4, the feast of Metrophanes of Constantinople.
Churches that follow the Old (Julian) Calendar keep the same feast on June 17 of the civil calendar. See everything commemorated on June 4.
The saint behind the name

Metrophanes of Constantinople
June 4, Hierarch, Μητροφάνης
St. Metrophanes was the first hierarch of Constantinople, shepherding the Church of Byzantium in the days when St. Constantine the Great made the city his capital. A man of peace and holiness, he guided his flock through the end of the persecutions into the freedom of the Church; by reason of his great age he could not travel to the First Ecumenical Council at Nicaea, but he upheld its faith.
Name days: Mitrofanis
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.