Averikios’s name day is October 22, the feast of Abercius of Hieropolis.
Churches that follow the Old (Julian) Calendar keep the same feast on November 4 of the civil calendar. See everything commemorated on October 22.
The saint behind the name

Abercius of Hieropolis
October 22, Hierarch, Αβέρκιος
St. Abercius was Bishop of Hierapolis in Phrygia in the second century, called Equal-to-the-Apostles for his missionary zeal. He overthrew the idols of his city, brought multitudes to faith in Christ by his preaching and healings, and journeyed as far as Rome and the lands of the East, everywhere confirming the churches.
Also celebrating on October 22
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.