Dometios’s name day is August 7, the feast of Dometios the Persian.
Churches that follow the Old (Julian) Calendar keep the same feast on August 20 of the civil calendar. See everything commemorated on August 7.
The saint behind the name

Dometios the Persian
August 7, Martyr, Δομέτιος
St. Dometios was a Persian of the fourth century who came to faith in Christ and left his homeland for the regions of Syria and Mesopotamia, where he received the monastic tonsure and later lived in ascetic solitude, drawing many who sought healing and counsel through his prayers.
Name days: Dometios
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.