I want to state that I personally believe Jehu not Hazael authored the Tel-Dan Stele.
The slaying of Jehoram son of Ahab King of Israel and Ahaziah son of Jehoram of the House of David are feats of Jehu not Hazael according to The Bible in II Kings 9 and 10 (and it’s mentioned in II Chronicles as well). And the Seventy kings or princes probably refers to the 70 sons of Ahab in Samaria. That of course doesn’t matter to people who want to say The Bible is wrong.
If the Author named himself, that part of the inscription has not survived. There are three maybe four main reasons people think it was Hazael.
First that generally Dan is not believed to have been under Israel’s control at this time, having been lost to Aram-Damascus back in the days of Baasha, but 2 Kings 10:29 contradicts the assumption that Jehu never controlled Dan, verses 32 and 33 seem to imply it was parts of modern Jordan that Hazael took from Jehu.
Second is that the author says Hadad was the Deity who made him King. Now at face value this is an issue with The Biblical account either way as First and Second Kings says that YHWH via Elijah and Elisha made both Hazael and Jehu King. But even though Jehu did continue the idolatry of Jeroboam, his name seems to imply he wouldn’t have attributed his kingship to any god but YHWH, while the Bible itself associates Hadad worship with the Kings of Aram via naming many (both before and after Hazeal) BenHadad and Hadadezer.
Thing is the Baal Worshiping Ahab and Jezebel still gave YHWH theophoric names to their children. If Baal was the name the Stele used that would not fit Jehu who was strictly suppressing Baal worship. But Zechariah 12’s reference to the “mourning of Hadadrimmon in the valley of Megiddo” suggests Hadad worship was also present in the very region where Jehu overthrew the House of Ahab. So Jehu may have been okay with using Hadad as an alternate name of YHWH.
Third is that the Author of the Stele seems to refer to earlier conflicts between the King of Israel and his father, and the Syrian Historian quoted by Josephus claims Hazael was a son of BenHadad. I think Hazael may well have publicly claimed to be a son of BenHadad in spite of how The Bible makes it seem pretty unlikely he actually was, but this is still not inherently inconsistent with Jehu. Jehu came from Gilead in the Trans-Jordan which may have often thought of itself as semi-independent. And the books of Kings repeatedly refers to there being more events then it records. Since Ramoth-Gilead was the site of the battle between Israel and Ben-Hadad, the father or grandfather of Jehu could have been an ally of BenHadad.
J-W Wesselius also argued the Stele was authored by Jehu.
