I do like the lottery idea. But I would start experimenting with mixed systems of professional politicians and laymen "juries". There are plenty of possibilities...

In general, I think that randomness can have many unrealized applications. For example, consider the following frequent voting dilemma: at an important election you would like to vote for a small party which represent your issues very closely, but for tactical reasons you have to consider voting for a big party as a lesser evil of the possible winners. You know the situation: if you vote for the small party, the worst winner may come out; if you vote for the big party, your issues will never be represented. What to do? Drop a coin in the voting booth! (Or a dice, if you wish to assign unequal weights for your intentions.) The point is that if many people will do this, the voting outcome will represent their average intentions very well, by the Central Limit Theorem! That would be much better than desperate persuasions and massive opinion swings, most often resulting in one of the two dreadful outcomes.

By the way, have you read "The Lottery in Babylon" by Jorge Luis Borges? I have to admit, I only started to read the story long ago.