Isaac’s name day is May 30, the feast of Isaac the Syrian.
Churches that follow the Old (Julian) Calendar keep the same feast on June 12 of the civil calendar. See everything commemorated on May 30.
The saint behind the name

Isaac the Syrian
May 30, Venerable, Ισαάκ ο Σύρος
St. Isaac the Syrian lived in the seventh century. A monk from his youth, he was consecrated Bishop of Nineveh, but after only a short time he laid aside the episcopate and withdrew into the desert, giving himself wholly to stillness, prayer, and the study of the Scriptures until blindness and old age overtook him.
Also celebrating on May 30
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.