Philemon’s name day is November 22, the feast of Philemon the Apostle.
Churches that follow the Old (Julian) Calendar keep the same feast on December 5 of the civil calendar. See everything commemorated on November 22.
The saint behind the name
Philemon the Apostle
November 22, Apostle, Φιλήμων
The holy apostle Philemon, one of the Seventy, was a Christian of Colossae and the recipient of the Apostle Paul's epistle that bears his name, in which Paul pleads for the runaway slave Onesimus as for a beloved brother. His household was a gathering place for the church of his city.
Also celebrating on November 22
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.