I know Yahtzee wasn't a fan of it (or really any JRPG, for that matter), but I think The World Ends With You did a good job with its difficulty thing.
Not only could you switch mid-game...or indeed, any time you wanted, you could also alter your level. Changing your difficulty (You start on Normal, gain Easy later, Hard after beating the story and you could buy Ultimate) changed enemy health and how much damage you recieved, and what kind of items they dropped. Lowering your level made them drop more and made you recieve more damage while dealing less, while upping your level did the opposite.
You could do this whenever you wanted, and it is in fact encouraged in game so that you can get different pins and items. Also, if you were a pussy and kept losing to a boss, you could replay the fight on Easy mode.
Unless your were fighting that elephant, because you HAD to beat it on Hard for it to count. Fucking purple elephant.