Certain games allow players to customize aspects of their character or in-game situation.  Consider Cosmic Encounter; each player has a special power representing their particular alien species.  Classically these powers are randomly selected, in the form of cards dealt out to the players at the beginning of each game.  Of course, some play groups might let players choose their alien races, or else negotiate the distribution.

That’s the great thing about powers on cards; you can easily shuffle them up and deal them randomly, or you can let players choose in some other method.  Those two methods would provide for different experiences; therefore, players have an easy way to vary their experiences of they choose.  The really important thing is that you’ve allowed players these two options without any extra work.

What other scenarios might profit from this strategy?  Some games allow customization, so let’s think of doing them randomly.  A randomly chosen character class in a game like Kingdom of Loathing?  How about a randomly generated Master of Orion 2 race?  How about the other side, a randomly generated option that might benefit from direct choice?  This one is actually much harder to think of; I suspect if people play a game where this happens, somebody eventually hits upon the idea of letting people choose; Diplomacy comes to mind.  I’d be happy to hear more examples in the comments!