There’s nothing better than getting a big, shiny medal or trophy for your in game accomplishments. There are two major ways I can see medals/trophies being used.
The first method is to award three medals at the end of every run. The three medals would be: time (how fast can you beat the level?), mobs defeated (can you defeat 100% of hostile mobs?), and map explored (can you find all the secrets and chests?). Some of these are currently in the game as an end of level stat, but nice, shiny medals would be much nicer to see and they could also award the player with better gear. These medals could have tiers as well: iron, gold, diamond (to stick to classic Minecraft ores).
The other use for medals could be for an indication of what the highest difficulty you completed a level on is. Seeing a little medal next to each level with a +20 on it to indicate all the levels I beat on +20 would be really pleasing to see. In addition, the player could have a medal next to their name for other players to see or in their hud (so the player can see) to show they completed every level on at least X difficulty.
I feel that both of these uses for medals will highly improve the rate of level replays for levels that don’t see a lot of play and add one one last challenge for the player.