Lucian’s name day is October 15, the feast of Lucian of Antioch.
Churches that follow the Old (Julian) Calendar keep the same feast on October 28 of the civil calendar. See everything commemorated on October 15.
The saint behind the name

Lucian of Antioch
October 15, Martyr, Λουκιανός
St. Lucian was a priest of Antioch, renowned as a scholar of the Holy Scriptures, who labored over the sacred text and taught many disciples in the school of Antioch. During the persecution at the beginning of the fourth century he was arrested and taken to Nicomedia, where after a bold defense of the faith he died in prison for Christ around the year 312.
Also celebrating on October 15
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.