Procopios’s name day is July 8, the feast of Procopios.
Churches that follow the Old (Julian) Calendar keep the same feast on July 21 of the civil calendar. See everything commemorated on July 8.
The saint behind the name

Procopios
July 8, Martyr, Προκόπιος
St. Procopios the Great Martyr suffered for Christ at Caesarea in Palestine during the persecution of Diocletian at the beginning of the fourth century. According to tradition he was first a persecutor of Christians, but was converted by a divine revelation, confessed Christ openly, and endured imprisonment and torture before being beheaded.
Also celebrating on July 8
These names share the feast, many of them other forms of the same name:
Prokopios, Prokopis, Procopius
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.