Eleutherius’s name day is December 15, the feast of Eleutherios.
Churches that follow the Old (Julian) Calendar keep the same feast on December 28 of the civil calendar. See everything commemorated on December 15.
The saint behind the name

Eleutherios
December 15, Hierarch, Ελευθέριος
The holy hieromartyr Eleutherios was a bishop of the early Church, raised in piety by his mother Anthia and consecrated to the episcopate while still young. According to tradition he suffered in the persecutions of the second century: brought before the emperor, he refused to deny Christ, endured many torments unharmed, and was finally put to death, his mother Anthia being slain beside him.
Name days: Eleutherios, Eleftherios, Lefteris, Lef, Eleftheris, Eleutherius, Eleftheria
Also celebrating on December 15
These names share the feast, many of them other forms of the same name:
Eleutherios, Eleftherios, Lefteris, Lef, Eleftheris, Eleftheria
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.