Pancratos’s name day is July 9, the feast of Pancratius of Tauromenion.
Churches that follow the Old (Julian) Calendar keep the same feast on July 22 of the civil calendar. See everything commemorated on July 9.
The saint behind the name

Pancratius of Tauromenion
July 9, Hierarch, Παγκράτιος
St. Pancratius was, according to tradition, a disciple of the apostles who was sent to Sicily and became the first bishop of Tauromenium, present-day Taormina. There he preached the Gospel, overturned the worship of idols, and gathered a flock for Christ in the earliest days of the Church.
Also celebrating on July 9
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.