Alexei’s name day is March 17, the feast of Alexios Man of God.
Churches that follow the Old (Julian) Calendar keep the same feast on March 30 of the civil calendar. See everything commemorated on March 17.
The saint behind the name

Alexios Man of God
March 17, Venerable, Αλέξιος
St. Alexios, the Man of God, was the son of a noble Roman family. On the night of his wedding he secretly left his home, choosing poverty for the sake of Christ, and lived for years as a beggar in the East, spending his days in fasting and prayer at the doors of the church.
Also celebrating on March 17
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.