Reni’s name day is May 5, the feast of Irene the Great Martyr.
Churches that follow the Old (Julian) Calendar keep the same feast on May 18 of the civil calendar. See everything commemorated on May 5.
The saint behind the name

Irene the Great Martyr
May 5, Martyr, Ειρήνη η Μεγαλομάρτυς
The Holy Great Martyr Irene lived in the early centuries of the Church. According to tradition she was the daughter of a pagan ruler and came to faith in Christ in her youth, receiving in baptism the name Irene, which means peace. Confessing the Lord openly, she endured many tortures with courage, and her endurance drew multitudes to belief.
Also celebrating on May 5
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.