Nestor’s name day is October 27, the feast of Nestor the Martyr.
Churches that follow the Old (Julian) Calendar keep the same feast on November 9 of the civil calendar. See everything commemorated on October 27.
The saint behind the name

Nestor the Martyr
October 27, Martyr, Νέστωρ
St. Nestor was a young Christian of Thessaloniki, a disciple of the Great Martyr Demetrios. When the gladiator Lyaios was slaying challengers to the delight of the Emperor Maximian, Nestor received the blessing of the imprisoned Demetrios, entered the arena with the prayer, God of Demetrios, help me, and overthrew the boastful champion.
Also celebrating on October 27
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.