Grzegorz’s name day is January 10, the feast of Gregory of Nyssa.
Churches that follow the Old (Julian) Calendar keep the same feast on January 23 of the civil calendar. See everything commemorated on January 10.
The saint behind the name

Gregory of Nyssa
January 10, Hierarch, Γρηγόριος Νύσσης
St. Gregory of Nyssa was the younger brother of St. Basil the Great, born in Cappadocia in the fourth century to a family rich in saints. As Bishop of Nyssa he defended the Nicene faith against Arianism, endured exile for the truth, and took part in the Second Ecumenical Council in Constantinople in 381.
Also celebrating on January 10
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.