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.
If you don't have any experience soldering and stuff, I really suggest going for a Wii portable, it's well documented, there's so many resources for you, plays gamecube games better than a Gamecube. they are fairly cheap, and well supplied, The PS2 is sorta like a 'new frontier' so it's less documented, and the only model that can be trimmed, is only one model of the ps2.