Cyrus’s name day is June 28, the feast of Sts. Cyrus & John the Unmercenaries.
Churches that follow the Old (Julian) Calendar keep the same feast on July 11 of the civil calendar. See everything commemorated on June 28.
The saint behind the name

Sts. Cyrus & John the Unmercenaries
June 28, Saint, Κύρος και Ιωάννης
On June 28 the Church commemorates the translation of the relics of the Holy Unmercenaries Cyrus and John. Cyrus was a physician of Alexandria who healed in the name of Christ without payment, and John a soldier who joined him; together they strengthened the confessors during the persecution of Diocletian and were beheaded for the faith at the beginning of the fourth century.
Also celebrating on June 28
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.