Haggai’s name means feast/festival,
which maybe indicates that he was born during one of Israel’s religious feasts, but this is simply a guess. Otherwise, we know nothing of him beyond what is found in the prophecy by his name. We do not know his age, or whether he was from a priestly line. The Holy Spirit did not consider these details important enough to share them with us.
1 In the second year of Darius the king, in the sixth month, on the first day of the month, the word of the LORD came by the hand of Haggai the prophet to Zerubbabel the son of Shealtiel, governor of Judah, and to Joshua the son of Jehozadak, the high priest: