I'd go with the Wii. It's super easy to get homebrew running on a Wii and only requires a compatible SD card (technically it's not required to actually install the Homebrew Channel, but you'll need one to actually run and store your emulators and ROMs). Just note that later produced consoles had the GameCube backwards compatibility removed.
Homebrew on a PS2 is more complicated. The easiest way to run homebrew is by purchasing a memory card with FreeMCBoot pre-installed. A better way (in my opinion) is to use a SATA modded Sony network adapter (it must be an original Sony unit) with a 1TB or less SSD (for compatibility reasons) and install PlayStation Broadband Navigator Definitive Edition. If you just want to use FMCB you'll still want a network adapter and drive, but you can use a third-party one that comes with SATA support out of the box. Another thing is that there were a lot of revisions to the PS2 and not all methods of accessing homebrew work with all models.
Edit: I forgot about the PS2 Slim. You can't use a network adapter with one, which means you can't use PSBBN without hardmods.
Edit 2: If you go with a Wii, then follow this guide to hack it.