Patrikios’s name day is May 19, the feast of Patrick, Bishop of Prusa.
Churches that follow the Old (Julian) Calendar keep the same feast on June 1 of the civil calendar. See everything commemorated on May 19.
The saint behind the name

Patrick, Bishop of Prusa
May 19, Hierarch, Πατρίκιος
The Holy Hieromartyr Patrick was Bishop of Prusa in Bithynia in the early centuries of the Church. Brought before the pagan governor with priests of his church, he confessed Christ as the Creator of all, refusing to ascribe the works of God to the idols, and for his boldness he was put to death by the sword together with his companions.
Also celebrating on May 19
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.