In an interview with VGC for The GameCubes's 20th anniversary, veteran Rare developer Martin Hollis revealed that not only was he among the first people to see "Project Dolphin", but also that he was possibly responsible for the GameCube's name and theme:

“I arrived in Kyoto, went into the big building, and Mr. Miyamoto and his team straight away took me to this empty meeting room and sat me down in front of a television [...] They switched it on, and Miyamoto told me to press the A button on the controller. I pressed it and the purple rolling cubes appeared on screen with the boot up music that we now know so well, revealing the GameCube name. [...] As the on-screen reveal happened, Mr. Miyamoto stared at my face intensely! That was my initiation, which was maybe because I’d actually suggested the name ‘Cube’ during my time at NTD. Months earlier I did a sheet of paper at Nintendo of America with a whole load of suggestions for names and one of them was ‘Star Cube’ or something like that.”

Nintendo did indeed trademark "Starcube" lending more legitimacy to Hollis' suspicion.
Contributed by PirateGoofy