Nino’s name day is January 14, the feast of Nina.
Churches that follow the Old (Julian) Calendar keep the same feast on January 27 of the civil calendar. See everything commemorated on January 14.
The saint behind the name

Nina
January 14, Holy Woman, Νίνα
St. Nina, Equal-to-the-Apostles, was a young woman of Cappadocia who in the fourth century brought the Gospel to the land of Georgia. Traveling there alone, she lived in simplicity and prayer, carrying a cross woven of vine branches, and through her preaching and her healings the king and queen came to believe, and the whole nation received holy baptism.
Also celebrating on January 14
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.