Gordius’s name day is January 3, the feast of Gordios.
Churches that follow the Old (Julian) Calendar keep the same feast on January 16 of the civil calendar. See everything commemorated on January 3.
The saint behind the name

Gordios
January 3, Martyr, Γόρδιος
St. Gordios was a centurion of Caesarea in Cappadocia who lived in the time of the persecutions of the early fourth century. Withdrawing to the wilderness to live in prayer, he returned to the city during a great pagan festival and openly confessed Christ before the assembled crowds, knowing the cost. He was condemned and beheaded for his confession.
Also celebrating on January 3
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.