Hermylos’s name day is January 13, the feast of Hermylos.
Churches that follow the Old (Julian) Calendar keep the same feast on January 26 of the civil calendar. See everything commemorated on January 13.
The saint behind the name

Hermylos
January 13, Martyr, Ερμύλος
St. Hermylos was a deacon who suffered for Christ under the Emperor Licinius in the early fourth century, in the region of Singidunum, the city known today as Belgrade. Refusing to deny his Lord, he was tortured and finally drowned in the waters of the Danube, together with his friend the jailer Stratonikos, who confessed Christ after witnessing his endurance.
Also celebrating on January 13
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.