Well, Jefferson offered Lafayette the governorship of Louisiana, but he demurred. I actually tried to start a discussion about this very same topic about a year ago. Now, one of the problems is that Lafayette was a staunch opponent to slavery. Even if he were to take the governorship and secured election after it attained statehood, a staunch anti-slavery proponent would have difficulty gaining the nomination of a natural party (now, he might have been able to blunt slavery in Louisiana, and buttress French culture in that state by inviting emigration of anti-Napoleon Frenchmen during that time). Now, a further problem is that the romanticism of Lafayette is based on the fact that he was French, was involved in French politics, and did not meddle in American affairs. The second he becomes an American politician, and was forced to take a stand on American political issues, that romanticism is going to die a quick and messy death.
Now, does that mean that he wouldn't still find a nomination due to his celebratory status? No. But it would be difficult to say the least. I think the best case would be a successful governorship, curtailing (probably not ending, at least at first) slavery in Louisiana, and a strengthening of French culture in that state. He is likely a loyal Democratic-Republican and I could see him taking command of US forces during the War of 1812. Maybe we see Lafayette becoming the hero who saved New Orleans. In that later case, a United States without Andrew Jackson is going to be a change from OTL. Not sure how politics would develop, but I could see the Lafayette family becoming entrenched in Louisiana and, maybe, his son being able to run for politics. But, as I said, a timeline without Andrew Jackson is going to be a huge change and I'm not sure how politics would develop (hell, maybe Lafayette becomes a staunch supporter of JQA. In a world with no Jackson, JQA might go down as one of the most successful presidents in the early Republic and, due to my own love of Adams, a utopia to be sure!

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