Onesimos’s name day is February 15, the feast of Onesimos.
Churches that follow the Old (Julian) Calendar keep the same feast on February 28 of the civil calendar. See everything commemorated on February 15.
The saint behind the name

Onesimos
February 15, Apostle, Ονήσιμος
St. Onesimos the Apostle was the slave of Philemon of Colossae who fled from his master and, by God's providence, came to St. Paul in his imprisonment. There he was converted, and Paul sent him back with the Epistle to Philemon, asking that he be received no longer as a slave but as a beloved brother in Christ.
Name days: Onesimos
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.