Baruch’s name day is September 28, the feast of Baruch the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on October 11 of the civil calendar. See everything commemorated on September 28.
The saint behind the name
Baruch the Prophet
September 28, Prophet, Βαρούχ
The holy Prophet Baruch, son of Neriah, was the faithful scribe and companion of the Prophet Jeremiah, writing down his prophecies and reading them before the people and the king. He shared the prophet's sufferings at the fall of Jerusalem and went with him into Egypt.
Also celebrating on September 28
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.