The Republic of Florence picked its ruling council by lot, from a pool of qualified (wealthy) men. This was supposed to be an incorruptible process, but lotteries were fixable if you were the Medici and had the money needed to bribe enough people.

Edit: a very readable source on the matter is http://www.exurbe.com/?p=37

This was done in ancient Greece; several democratic states chose their legislature by lot. Are you asking for modern equivalents of this practice?

The copts, i.e. the orthodox church of Egypt, choose their pope by lot. The synod of bishops and other dignitaries compiles a list of three candidates by elections. The names are put into a chalice and one is drawn.

It's called demarchy.