I've been really wondering this myself. One of my TL ideas has Jefferson as an early central figure and I considered just this.
What comes immediately to mind:
French Revolution/Quasi-war/XYZ: Jeffy is a Francophile and OTL stubbornly refused to see the grim reality of Robespierre and the FR in favor of continued delusions that it mirrored the ARW. How does he react to the RevFrance war with Britain? Does he openly support the French? Does this butterfly the QW? Does the US end up in "QW" with Britain instead?
Alien/Sedition Act: butterflied...but what precedents are set in the press?
National Bank: DOA. How does he deal with finance?