Yanni’s name day is January 7, the feast of John the Baptist.
Churches that follow the Old (Julian) Calendar keep the same feast on January 20 of the civil calendar. See everything commemorated on January 7.
The saint behind the name

John the Baptist
January 7, Great Feast, Ιωάννης ο Πρόδρομος
On the day after Theophany, the Church gathers to honor St. John the Forerunner and Baptist, the greatest of the prophets, who was found worthy to baptize the Lord in the Jordan. Son of the priest Zacharias and the righteous Elizabeth, John lived in the wilderness in fasting and prayer, preaching repentance and preparing the way of the Lord.
Name days: Ioannis, Giannis, Yiannis, Yannis, Yanni, Giovanni, John, Jean, Juan, Ivan, Vanya, Yianni, Gianni, Giannakis, Prodromos, Johnny, Ioanna, Gianna, Yianna, Giannoula, Joanna, Joanne, Joan
Also celebrating on January 7
These names share the feast, many of them other forms of the same name:
Ioannis, Giannis, Yiannis, Yannis, Giovanni, John, Jean, Juan, Ivan, Vanya, Yianni, Gianni, Giannakis, Prodromos, Johnny, Ioanna, Gianna, Yianna, Giannoula, Joanna, Joanne, Joan
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.