GameCube - Did You Know Gaming? Feat. SpaceHamster
Pikmin - Did You Know Gaming? Feat. JonTron
Nintendo had once patented and developed a prototype motion controller for the GameCube; this controller went so far into development that game developer Factor 5 experimented with them for potential use in their launch titles.
Contributed by game4brains
There exists a GameCube Service Disk that was used by the Nintendo World Class Service to test and diagnose problems with the console and peripherals. It is assumed that once Nintendo retired the World Class Service, they took back the disks and destroyed them, which would explain why there are very few ever found.

This disk contains a fair amount of unused content, from unused sounds and unused test pictures to stereoscopic 3D test pictures.
Contributed by SonicManEXE
The Panasonic Q was a version of the Nintendo GameCube that had the ability to play DVDs, audio CDs, and MP3s. The console ceased production after two years and only sold 100,000 units. One cause of the console's failure was that it was actually cheaper to buy a GameCube and a DVD Player separately.
Contributed by Pogue-Mahone
The GameCube controller's design possibly originated from a prototype controller for the Virtual Boy. They were both purple, and had a similar shape.
Contributed by Boyobmas
The GameCube is actually capable of rendering stereoscopic 3D. The GameCube was going to have an add-on that would attach to a TV and enable it to properly display 3D games. The first planned game for the add-on was Luigi's Mansion. The add-on was eventually cancelled as it would have cost more than the console itself.
Contributed by gamerofgames
Nintendo ran a contest to help promote the upcoming release of the GameCube called "What would YOU do for a Nintendo GameCube?" where fans were picked to perform stunts that they'd chosen themselves to win a GameCube, Game Boy Advance, a games package, and USD. Some of the acts which were picked by entrants were to paint Nintendo logos with their tongue, juggling three consoles whilst dressed as a game character, painting and shaving themselves to be a Pikmin, and eating a GameCube made of spam, chocolate syrup, and cat food.
Contributed by KnowledgeBase
Two of Nintendo's biggest franchises, Mario and The Legend of Zelda, have water-centric games on the Gamecube. Some have speculated this is somehow related to the GameCube's early project name "Dolphin".
When the GameCube was released, Nintendo targeted a 50 million sale goal by 2005, in order to compete with Sony and Microsoft. The consoles ended up only reaching 21.74 million units sold (estimate).
Contributed by ThisGuyInTheSuit
The GameCube's SDK has strings that reference all of the known N64 peripherals, including obscure ones such as the keyboard and mouse, suggesting that the GameCube was once planned to have support for Nintendo 64 peripherals.
Contributed by xianc78
Before it was released, a tech demo for the Gamecube was developed which featured an explorable version of Princess Peach's castle. This demo was later leaked online.
Contributed by DokemonStudios
The working title for the GameCube was the Dolphin. As a result, many games from the GameCube era reference this, such as Super Mario Sunshine's setting being in the shape of a dolphin (the island is also named Isle Delfino; "Delfino" is Italian for "Dolphin") and Captain Olimar's ship in Pikmin being named the S.S. Dolphin. The GPU of the machine is named "Flipper", another reference to the console's codename.
Contributed by lividd3ad
The soft reflections used for the GameCube's startup animation and menu are the same texture file that Ocarina of Time and Majora's Mask use for shiny items.
Contributed by SonicManEXE
There are two hidden alternate start-up sounds which can play after you power-up the console. The first one features a squeaking sound and a child's laughter, which plays when you have a controller in port 1 with the "Z" button hold down as you power-up. The second is of Japanese instruments played which activates via the same method but instead with four controllers.
Contributed by natogamer583
There is a microscopic image of a Dolphin within a computer chip inside the GameCube, referencing the GameCube's codename during development.
Contributed by Dazz
The GameCube's BIOS menu has slow, seemingly random ambient background music.

This background music, when sped up to sixteen times its normal speed, is actually the intro jingle for the Famicom Disk System, a Famicom add-on released by Nintendo in 1986.
Contributed by Vipershark