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

Tychon of Amathus
June 16, Hierarch, Τύχων
St. Tychon was Bishop of Amathus in Cyprus in the fifth century. The son of a baker, he was known from youth for his generosity to the poor, and as bishop he labored with apostolic zeal to turn the remaining pagans of the island from their idols to Christ, shepherding his flock with gentleness and prayer.
Also celebrating on June 16
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.