Sebastian’s name day is December 18, the feast of Sebastian.
Churches that follow the Old (Julian) Calendar keep the same feast on December 31 of the civil calendar. See everything commemorated on December 18.
The saint behind the name

Sebastian
December 18, Martyr, Σεβαστιανός
The holy martyr Sebastian was a Roman soldier of high rank who suffered under the persecution of Diocletian at the end of the third century. A secret Christian who strengthened imprisoned confessors, he was denounced, bound to a tree, and shot with arrows; found still living, he confessed Christ again before the emperor and was then put to death. A company of martyrs who suffered with him is honored alongside him.
Name days: Sebastianos, Sebastian, Bastian
Also celebrating on December 18
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.