I've had this idea for quite some time - I'm going to copy and paste some of my ideas for controllable stages:
Stage: Wily's Fortress
Controllable Element: Yellow Devil (working under the assumption that he is not actually a boss)
How to Control: Yellow Devil will begin on the left side of the screen. Tapping a direction on either control stick will cause a chunk of Yellow devil to fly across the screen, which must be dodged by players or else they will be damaged. Tapping A or B will cause Yellow Devil to slam down with his hand, temporarily trapping and damaging those caught under his fist.
Stage: Pyrosphere
Controllable Element: RIDLEY
How to Control: Ridley will be in the background of the stage. Ridley can be slowly controlled using the two control sticks. Tapping A or B will cause Ridley to shoot a ball of plasma, which travels towards the foreground, damaging any players. Tapping X or Y, on the other hand, will cause Ridley to swoop forward in his signature fashion and grab on to a player. Both of these moves have meters that must fill up before the maneuver can be used again - simply so it isn't annoyingly overpowered.
Stage: Nintendogs
Controllable Element: The puppy
How to Control: You can interact with the dog with the touch screen - such as petting it or even placing treats around the stage! Treats cause the dog to move between the background and the foreground, causing some of the blocks to be rearranged.
And a couple for my own stage ideas...
Stage: Luigi's Dreams
Controllable Element: Luigi(?)
How to Control: This stage comes from Mario & Luigi: Dream Team! It takes place inside Luigi's dream world. A sleeping Luigi will be displayed on the gamepad screen (but NOT as a part of the stage itself). Like in M&L:DT, interacting with the snoozing Luigi will cause things to change in the Dream World.
Stage: Toad's House
Controllable Element: Toad
How to Control: This stage will be a simple one, taking place inside and on top of the Toad house from many Mario platformers. Toad will be standing in his usual spot. The gamepad's touch screen will have a huge grid with all of the items visible - tapping them will cause Toad to open a chest containing that item. This could definitely create some fun and frantic matches.
Obviously, there would be an option to set a CPU to control these hazards or even to disable them entirely.