Never GameCube. Wiis are easier and cheaper to portablize and come with a host of better features, including easy to get 480p video, more emulation options, and they can do everything a GameCube can, barring a few obscure peripherals that no one actually cared about. There's a whole bunch of other reasons too, but the TLDR is that Wii is superior 100%, with the exception of nostalgia for some.
As for Wii VS PS2, it's really up to you. Wii portablizing is marginally more refined right now with things like GC+, but honestly the level of difficulty between making a PS2 or a Wii portable isn't drastic. The PS2 will get better better life for sure, but again, that's not a deal breaker. If the ability to play GC/Wii games on the go appeals to you more, then build a Wii portable. If the PS2 was your console of choice growing up, then make a portable PS2. Either way, there are plenty of guides and experts here for you to have the info you need to make one.