I was also thinking of the when the next console is released, will the Wii U Touchscreen Tablet of Doom end up being replaced or allowed to work? It'll then need a name of its own, and "New controller" isn't going to work anymore. (New Super Mario Bros. isn't so New anymore).
Right now, we don't know if the old Wii remotes, nunchucks, or any of the classic controllers will be bundled with the console or not. If the new controller is ever to be sold in the future, what would it be called (these things will break), and i doubt that most of the consumer population will call Nintendo for every thing that needs to be repaired or replaced.
Anyway, for the games, there'll be instructions to how to play them, and they're gonna have to call the controller something, right?
You know when you start a Wii game that requires the Nunchuck, like Super Mario Galaxy or Metroid Prime Trilogy, it'll let you know. Say that your tablet screen wasn't turned on, sync'd or wasn't charged up (dead battery), the game may ask that you please connect the "_____ controller" to the Wii. "New Controller" shouldn't be it. If the device doesn't have a name, then it'll only confuse some people, unless it's going to be the "System Name Controller", like the Gamecube, N64, SNES, and NES. But the difference here is that, like the Wii, this new controller offers a unique way of playing on home consoles, and will be used with a bunch of other Wii controllers for multiplayer and Virtua Console gaming.
Just because it comes with the console doesn't mean that it'll always be the controller that consumers will use as the default controller. I'm sure that Nintendo will come up with something simple and easy for most consumers to understand.