Controller function
The idea of adding vibration feedback to controllers was first introduced by the Nintendo 64 console in 1997, following the release of Star Fox 64. The peripheral, called the "Rumble Pak", connected to the expansion port built into the Nintendo 64 controllers, relied on a motor coupled with an unbalanced weight; whenever the motor spun its axle, the weight would cause rapid shifts in inertia in the controller, leading to the vibrating mechanism.
Initially expected to be a novelty, the "Rumble Pak" later spread, with many games later supporting the accessory, and some older games were actually re-released in order to take advantage of the peripheral. Among the games to feature Rumble Pak compatibility was Super Smash Bros.; the game would make the controllers rumble whenever the player's character was hit or performing an attack, amongst other events.
The later GameCube and Wii consoles both had controllers that would vibrate, but in these cases, the mechanism was built into the controller, as opposed to the Nintendo 64. The Classic Controller and Nunchuk, however, lack rumble mechanisms, though the Wii Remote they are connected to will still vibrate. In both games, the controller rumbles slightly more often, such as before the Results screen and before starting matches in the Classic Mode.
 Menu setting
Melee and Brawl both have "Rumble" settings in the Options menu, where the player can select whether or not they want their controllers to vibrate or not.
Individual names also can have Rumble features on or off; in these cases, selecting the name overrides what the player's controller port is set to.
As the Rumble Pak can simply be removed from the Nintendo 64 controller, there is no Rumble menu in the original game.