I'm going to have to echo the idea that the most important part is the games. You can get NASCAR Thunder 2005 for any of the three systems, since your family seems to like that style of game.
PS2 seems to have the widest variety of racing games, but X-Box is close behind. For both of those systems, the games tend to be in more realistic settings: stock car, rally car, underground street racing, etc. For GameCube, some of those type exist, but the focus is more on gameplay than realism. There's MarioKart: Double Dash, which is excellent fun multiplayer, and F-Zero GX which is a futuristic racing game that's also quite fun.
That's not to say that each system
doesn't have games of the other type, it's just what tends to be the focus.
I would recommend a GameCube, if only for the exclusive titles on it. Like mentioned before, it has a wider base of games that your child would enjoy, and many that your husband would as well. As far as racing games go, I recommend the two I mentioned before, as well as Need for Speed Underground 2, along with the NASCAR game (not really my cup of tea, but it scored well, so I'm sure it's a pretty good game). You can then get a third party steering wheel, and you'll be all set. Not to mention that a GameCube costs about half what the other systems do.
Eh, just my two cents.
Edit: I just took a look at EBgames.com and saw that they have a GameCube Holiday bundle that comes with a Platinum Gamecube, 1 Controller, and Mario Kart: Double Dash for $100. A good deal by all standards, and maybe something that you could take a look at. Find it
here.