Hosea’s name day is October 17, the feast of Osee the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on October 30 of the civil calendar. See everything commemorated on October 17.
The saint behind the name

Osee the Prophet
October 17, Prophet, Ωσηέ
The holy Prophet Hosea was one of the twelve minor prophets, who prophesied in the northern kingdom of Israel in the eighth century before Christ. Through the image of his own sorrowful marriage he proclaimed the unfailing love of God for His unfaithful people, calling Israel to repentance with words of both judgment and tenderness.
Also celebrating on October 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.