Sozon’s name day is September 7, the feast of Sozont.
Churches that follow the Old (Julian) Calendar keep the same feast on September 20 of the civil calendar. See everything commemorated on September 7.
The saint behind the name

Sozont
September 7, Martyr, Σώζων
St. Sozon was a shepherd of Cilicia in Asia Minor who tended his flock in prayer and meditation on the Scriptures. During a time of persecution he confessed Christ boldly at Pompeiopolis, and after enduring torture he gave up his soul to God, receiving the crown of martyrdom.
Also celebrating on September 7
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.