The government is a random sample of the population. I suspect it wouldn't actually be that different, but someone ought to try it.

Sortition. Famously used in Athens.

If we're going by weird procedures, I like the ones used for electing the Doge of Venice:

New regulations for the elections of the doge introduced in 1268 remained in force until the end of the republic in 1797. Their object was to minimize as far as possible the influence of individual great families, and this was effected by a complex elective machinery. Thirty members of the Great Council, chosen by lot, were reduced by lot to nine; the nine chose forty and the forty were reduced by lot to twelve, who chose twenty-five. The twenty-five were reduced by lot to nine and the nine elected forty-five. Then the forty-five were once more reduced by lot to eleven, and the eleven finally chose the forty-one who actually elected the doge.