Babylas’s name day is September 4, the feast of Babylas of Antioch.
Churches that follow the Old (Julian) Calendar keep the same feast on September 17 of the civil calendar. See everything commemorated on September 4.
The saint behind the name

Babylas of Antioch
September 4, Hierarch, Βαβύλας
St. Babylas was Bishop of Antioch in the third century, remembered for his boldness in barring an emperor stained with blood from entering the church until he should repent. During the persecution under Decius he was imprisoned for his confession of Christ and died in chains around the year 253.
Name days: Babylas
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.