Athenogenes’s name day is July 16, the feast of Athenogenes.
Churches that follow the Old (Julian) Calendar keep the same feast on July 29 of the civil calendar. See everything commemorated on July 16.
The saint behind the name

Athenogenes
July 16, Hierarch, Αθηνογένης
St. Athenogenes was a bishop in the region of Sebaste who suffered martyrdom with a company of his disciples during the persecutions at the turn of the fourth century. Refusing to deny Christ, he strengthened his spiritual children as they went before him to death, and then received the crown of martyrdom himself.
Name days: Athenogenes
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.