Panteleimon’s name day is July 27, the feast of Panteleimon the Unmercenary.
Churches that follow the Old (Julian) Calendar keep the same feast on August 9 of the civil calendar. See everything commemorated on July 27.
The saint behind the name

Panteleimon the Unmercenary
July 27, Martyr, Παντελεήμων
St. Panteleimon the Great Martyr and Healer was born in Nicomedia in the late third century. Trained as a physician, he was brought to Christ by the priest St. Hermolaos, and thereafter healed the sick freely in the name of the Lord, taking no payment, for which he is counted among the Holy Unmercenaries. Denounced during the persecution of Maximian, he endured many tortures and was beheaded around the year 305.
Name days: Panteleimon, Pantelis, Panteli, Telis, Pantaleon
Also celebrating on July 27
These names share the feast, many of them other forms of the same name:
Pantelis, Panteli, Telis, Pantaleon
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.