Beating a certain level a certain number of times unlocks a new hat for your StreetPass Mii.

Hat	How to unlock
Arwing Hat	Beat Restless Rise - South (Room 5).
Beehive Wig	Beat Throne Room for the second time.
Boo Hat	Beat the first room, third playthrough.
Cheeseburger Hat	Beat Granite Golem - (Room 8, Top Path).
Dark Lord Helmet	Defeat the Dark Lord three times, having every room in find mii 2's regular quest cleared.
Donkey Kong Hat	Defeat Armored Demon - (Castle of Darkness - second to last room).
Epona Hat	Beat Roaming Mummy and Marauding Mummy - (Castle of Darkness - Courtyard).
Floral Hat	Beat Forest of Sorrow Exit.
Lacy Headband	Beat the Dolorous Deepwood - West in the special quest, middle path.
Ninja Hood	Beat Diabolic Alter - East (Room 7, Bottom Path).
Panda Hat	Clear the second stage of Find Mii 2's special quest, middle path (Ransacked Ruins - West).
Peach's Crown	Clear the first room in Find Mii 2's regular quest a second time.
Pirate Hat	Clear the third room along the bottom path of Find Mii 2.
Pizza Hat	First room in the castle of darkness, third playthrough.
Prince's Crown	Beat Dark Lord for the first time.
Shyguy Mask	Beat the locked room in the regular quest.
Soccer	Beat room before the locked room.
Tomato Hat	Beat the Blood Ghost - (Merciless Mountain - West).
Top Hat	Beat Bewitching Bog - East (Room 3, Top Path).
Triforce Hat	Clear the fourth room on the second playthrough of Find Mii 2.
Wario Hat	Defeat the Dark Lord twice.
Warp Pipe Hat	Clear the first room in Find Mii 2's special quest for the first time.
Yoshi Hat	Beat the first room.