Simon’s name day is May 10, the feast of Simon the Zealot.
Churches that follow the Old (Julian) Calendar keep the same feast on May 23 of the civil calendar. See everything commemorated on May 10.
The saint behind the name

Simon the Zealot
May 10, Apostle, Σίμων ο Ζηλωτής
The Holy Apostle Simon the Zealot was one of the Twelve chosen by the Lord. The Gospels record his name among the Apostles, and his title of Zealot bespeaks his burning devotion. After Pentecost he went forth like the others to preach the Gospel, and tradition holds that he carried the good news to distant lands and completed his course as a martyr.
Also celebrating on May 10
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.