I was thinking about how, when the US had its constitutional convention, there was a big punch-up over federalists vs. anti-federalists to do with the power of the federal government. Also it was considered more acceptable to do indirect elections then, like how some states contributed their electoral votes to the President by the state legislature choosing rather than the voters. So I had this idea for a looser, more confederal setup for the executive if the anti-federalists had won out a bit more and the federal government was weaker. It's inspired partly by the EU's rotating presidency system and partly by Papal conclaves.
Basically, the Governors of the (then 13) states all get together in a 'conclave' and have to pick one of their own number to serve as National Governor (instead of President). This would carry a fairly short term, say one or two years, and consecutive terms would be forbidden (although not necessary non-consecutive multiple terms). While the Governor in question was acting as National Governor his lieutenant would take over his normal duties in his own state.
The small states would probably want every Governor to have an equal vote (like EU commissioners) but I can't see the big states allowing for that, so the 'conclave' would probably work similar to the OTL electoral college, except that instead of a state's electoral vote number literally corresponding to that number of electors, each Governor would cast a number of votes equal to the state's electoral vote.
Now this system would probably work quite well when the States were fairly nonpartisan, but it's interesting to speculate what would happen when political parties came in...