The Gamecube (with an add-on) can be hooked up with ethernet.
GC-G/L (http://www.gc-linux.org/wiki/Main_Page) is the GNU/Linux I'm running
on my GameCube.
In this case, firmware isn't a privacy issue-- it's a freedom issue.
Even if you hook up your GC to the internet, it's very unlikely there is any
sort of "phone-home" backdoor in it's firmware.
The GC has non-free firmware, but if you want an old console with GNU/Linux,
it's probably the best.
The better option, of course, would be to just hook a PC up to your TV-- a
nice console-like one is the TP Wee Penguin.