(For reference, I own both a PS2 and an Xbox... And I recently sold my Gamecube. I say this just to show that I'm not some sort of fanboy, and have fully experienced all three consoles.)
As a matter of fact, the Xbox is by far my favorite console, but I'm going to recommend the PS2.
Sony's PS2 has the broadest range of titles. (By contrast, the Gamecube has the narrowest.) With the PS2, it would not be difficult at all to find a game for each member of your family.
Also, it has a DVD player, which would be so much more convenient than using your PC... Probably higher quality, too.