We have all the current systems as well as most of the prior ones. I think the only 2 exceptions for us would have been the Dreamcast and the Saturn. Other than these though, we've had them all.
IMO, there is no one best system. If you want to talk hardware only, the Xbox wins hands down for the generation you're talking about. The Gamecube comes in second and the PS2 is last. Hardware alone however means little if the system doesn't have the games you prefer and in all honesty, all 3 systems have much to offer depending on your personal preferences.
For starters, you want DDR. I don't think DDR is available on the Gamecube. Nintendo did release their own version of the game though, but I can't remember what it's called.
The PS2 is your ONLY option really if your a fan of the Japanese style RPGs. There is little to no competition in this area on the other systems.
Xbox is hands down the best system for shooter games. They win hands down in the 1st person genre as well as the 3rd person. IMO, the other systems can't even come close in this area.
While it's only my opinion (and not everyone will agree), I feel that the Gamecube is one that never forgets the fun factor. I don't feel there are as many games for the Gamecube as the other 2, there are MANY excellent titles to choose from.
All 3 systems (with a less edge on the Gamecube) offer excellent racing and sports type games. You simply can't go wrong with any of them for these types of games though.
The 2 benefits in hardware that are worth pointing out though are:
Xbox - has a built in hard drive, so no need to buy memory cards - can plug in 4 controllers from the box - is Live ready out of the box (modem). There are other benefits too, so this isn't an all inclusive list, but a list of features worth pointing out.
Gamecube - The Wavebird wireless controller is hands down the best wireless controller on the market and those batteries last forever. The cost of one is the same as a regular controller for the Xbox too. The PS2's controller I believe is $5.00 less (this assumes no sale on any).
In our house, the Xbox gets the most action as we're big fans of the shooting genre. The Gamecube comes in 2nd and while we don't have as many games, the ones we do have are excellent and can't be bought for any other system. If I had to do it again, I wouldn't buy the PS2. There is little available on it that can't be had on the other systems too and we're not fans of RPG (role playing) games whether they're the Japanese type of the American (found more on Xbox) type.
Just don't listen to anyone who tells you one is better than the other. It will break down to what your family is looking for that matters the most.