Hermogens’s name day is December 10, the feast of Menas, Hermogenes and Eugraphus.
Churches that follow the Old (Julian) Calendar keep the same feast on December 23 of the civil calendar. See everything commemorated on December 10.
The saint behind the name

Menas, Hermogenes and Eugraphus
December 10, Martyr, Μηνάς
The holy martyrs Menas, Hermogenes, and Eugraphus suffered at Alexandria in the early fourth century. Menas, called the Melodious for his eloquence, was sent to the city on imperial business and openly preached Christ there; Hermogenes, sent to judge him, was won to the faith by his endurance and the grace shown in him, and Eugraphus, the companion of Menas, confessed Christ with them. All three were put to death by beheading.
Also celebrating on December 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.