Ilarionas’s name day is October 21, the feast of Hilarion the Great.
Churches that follow the Old (Julian) Calendar keep the same feast on November 3 of the civil calendar. See everything commemorated on October 21.
The saint behind the name

Hilarion the Great
October 21, Venerable, Ιλαρίων
St. Hilarion the Great was born near Gaza in Palestine at the end of the third century. Inspired by the example of St. Anthony the Great, whom he visited in the Egyptian desert, he took up the ascetic life in his homeland and became the father of monasticism in Palestine, drawing many disciples by his fasting, humility, and gift of wonderworking.
Also celebrating on October 21
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.