I've never heard this theory offered, and in any case it would seem to fail a very basic analysis. The bullet that was lodged in Jackson's chest was the result of his duel with Dickinson in 1806, an episode which can stand as evidence of his angry and violent personality. The duel is especially indicative of this, as it was provoked, many suspect, by Jackson's dislike of Dickinson over an insult given to Jackson's wife (officially it was a dispute that had arisen around race horses), and Jackson received no small amount of public furor for his behavior in the duel, when he initially withheld his fire, allowing Dickinson to shoot him, in order to not be rushed in aiming, so as to ensure that he would hit - and kill - his opponent. The quote attributed to him afterwards is variously given as "I could have been shot in the head, and I still would have killed him!"

And while this is the most famous of his encounters, there were others before it, some much more hotheaded than even his encounter with Dickinson. For instance, he found himself in a dispute with Gov. Sevier, which produced multiple outbursts of violence, or near to it, over several years, including both drawing pistols on each other in the street and exchanging fire in 1799, and a few years later a more formal challenge to duel that was adjudicated before the meeting.

So the simple fact is that Jackson's reputation for anger and violence predates the 'lead ball', and we have examples from his biography which attest to it as well. It isn't impossible that it didn't help exacerbate an already volatile and touchy personality, but it is doubtful there is really any way to test such an idea empirically.

u/Georgy_k_Zhukov has a much better answer below. But I love medical rabbit holes. From a medical perspective, is it plausible? Absolutely! First, we have a case series from 1983 of "plumbism" (a somewhat antiquated name for lead toxicity) from retained bullet fragments. The opinion of the reviewers was that the effect of lead toxicity was highest when the bullet was lodged in joint spaces (not in a large cavity like the chest). The cases in this article are fascinating and presented dramatically -- with anemia, neurologic sequela, and personality changes, and in all cases were quite disabling.

These sorts of cases series prompted the CDC to evaluate serum blood levels on a population basis in patients with retained bullet fragments (based on a state database). They found that in retained bullet fragment cases, elevations of blood lead levels were common, and often had nonspecific complaints like fatigue, abdominal pain, and memory loss. Looking from the other direction, a retained bullet fragment was an incredibly uncommon cause of elevated blood levels, less than 0.3%. They also performed another case series that identified less than 100 cases.

Ultimately, the CDC wasn't able to offer much of a recommendation, other than that patients with unexplained elevated blood lead levels should be asked about retained bullet fragments.

So ... is it possible? Of course. But I'm on #team u/Georgy_k_Zhukov on this one. The fact that this was a pre-existing personality trait, that the bullet was lodged in a large cavity, and that apparently he didn't suffer any other effects of lead toxicity, all suggest that plumbism probably wasn't playing a huge part.