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

Aggaeus the Prophet
December 16, Prophet, Αγγαίος
The holy prophet Aggaeus, called Haggai in the Hebrew Scriptures, is one of the twelve minor prophets. He prophesied in Jerusalem after the return from the Babylonian captivity, stirring up the people and their leaders to rebuild the Temple of the Lord, and declaring that the glory of the latter house would surpass the former.
Also celebrating on December 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.