Google Alerts found the following proposal and discussion, going much along the standard lines.
The Athenian magistrate system had many problems during it’s long life, and one of them was the issue of rule via oligarchy: in a democratic system driven by voter elections (as championed by Socrates), magistrates could effectively buy their seats. In turn, the Greek administration created by popular vote came represent only the interests of the wealthy.
This problem was solved by discarding elections in favor of sortition – simply drawing from the public at random whom would hold what seat for a given term.
I think this is a system that ought to be seriously considered for use today (of course it will never be, but whatever, I can pretend that this is totes up for debate somehow & somewhere).
…So you just pick people at random, and boom, there’s your government?
In essence, yes. There’s an annual lottery (or bi-annual, or however many times over ‘X’ time period you want to rotate people out), and everyone that meets eligibility criteria (so, presumably, no children) are included in said lottery. If your name / SSN / whatever is drawn, you fill a seat. It’s a paid position, just like today, and you otherwise do exactly what government officials do (or are supposed to do) right now.