Eudokimos’s name day is July 31, the feast of Eudokimos the Righteous.
Churches that follow the Old (Julian) Calendar keep the same feast on August 13 of the civil calendar. See everything commemorated on July 31.
The saint behind the name

Eudokimos the Righteous
July 31, Venerable, Ευδόκιμος
St. Eudokimos the Righteous was born in Cappadocia in the ninth century to devout parents. Serving as an officer in the imperial service, he lived in the midst of the world a life of hidden holiness, guarding purity, giving alms generously to the poor and to widows and orphans, and avoiding all display, with a special reverence for the Theotokos.
Also celebrating on July 31
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.