The fatal duel took place on the Weehawken dueling grounds, across the river from Manhattan, the same place where Hamilton's son had been killed mere years prior. (Please correct me if I'm wrong.)
Burr was known to be an excellent marksman, but Hamilton both (1) made known that he intended to miss the first shot, and (2) took every step to provoke Burr, from wearing his glasses to sighting the pistol before the duel.
These seem like contradictory acts -- either Hamilton intended to throw away his shot, or he wanted to make sure he had the best chance of firing an accurate shot.
One theory reconciles all these intentional acts: Hamilton intended to martyr himself on the spot where his son was killed.
He told friends that he had no intention to kill Burr to preserve his own legacy. He feigned concern for his own aim so Burr would take a well-aimed shot himself. And he picked a meaningful place to die.
Is there any merit to this theory, or any flaw in my assumptions? (For example, if there was no other dueling ground, the choice of Weehawken becomes less obviously intentional.)
Of possible interest here for considering Hamilton's motivation is the pistols themself, which Hamilton had selected and borrowed from his brother-in-law, John Church. Both Hamilton and Burr had a prior connection to the pistols; Burr had faced John Church in a duel 5 years prior and had a button shot off his coat by one of the set, and they were the same set of pistols that had killed Hamilton's son.
Besides the pistols' unusual history, they also had a number of features not commonly found on dueling pistols of the time, including a weighted barrel, adjustable sights, and a secret hair-trigger setting that would change the pistol from ~12 pound pull to a half pound or so. Hamilton likely knew about this secret from his brother-in-law prior to borrowing the pistols; Burr probably did not get the full feature tour during his prior encounter with the pistols and Church.
Duelists who intended to miss their target would typically fire directly into the ground, not over their opponent's head while lowering to aim. I believe the only source we have for Hamilton stating he intended to miss his shot was his second, who obviously is not an unbiased observer.
So, perhaps Hamilton was seeking a poetic martyrdom, and chose the pistols because of their connection to his family and animosity towards Burr. Or perhaps he chose them with the intent of having an advantage over Burr by setting the hair trigger, only to find he accidentally fired prematurely in the heat of the moment.
Source: This old Smithsonian article: http://www.aaronburrassociation.org/Smithsonian.htm