If I'm not mistaken, the "official" abbreviation for the Gameboy Advance was AGB, although most people who posted about it around here, at least, referred to it as the GBA (with good reason). Maybe Nintendo just wants to be different, though I can't help but think that they'd have been much better off leaving the "GCN" for their online service - GameCube Network, anyone?
NGC sounds like NGPC, which means Neo-Geo Pocket Color. Its too similar.
I like GCN. It sounds better than NGC.
PSX actually has a story about why its PSX and not PS or PSOne (PSOne is different from PSX or PS1):
The Play Station was originally being developed by Sony and used SNES parts and would also play SNES games. Nintendo pulled out of the deal, Sony got mad, and decided to continue with the Play Station. Later, they announced it as a 32-bit console and called it PlayStation X. It eventually became PlayStation, but PSX stuck with it.