Dionisis’s name day is October 3, the feast of Dionysios the Areopagite.
Churches that follow the Old (Julian) Calendar keep the same feast on October 16 of the civil calendar. See everything commemorated on October 3.
The saint behind the name

Dionysios the Areopagite
October 3, Hierarch, Διονύσιος Αρεοπαγίτης
St. Dionysios the Areopagite was an Athenian of the first century, a member of the court of the Areopagus, converted when the Apostle Paul preached in Athens of the God whom the city honored without knowing. According to tradition he became the first Bishop of Athens and ended his life as a martyr.
Name days: Dionysios, Dionysis, Denis, Dionisis, Dionysius, Dennis
Also celebrating on October 3
These names share the feast, many of them other forms of the same name:
Dionysios, Dionysis, Denis, Dionysius, Dennis
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.