If they don't mind paying outlandish prices for what is sub-standard gaming, get an XBox. Definitely the better of the two. Superb games, and even the original releases surpass most PS2 games and can be bought for £20. And then there's the option of XBox Live!, a mutliplayer gaming network which is apparently the bees knees. It's fast emerging to be the PS2 beater.
But if it's quality of games you want, can I suggest a retro console? A Dreamcast or Saturn would be fantastic, games are as low as £3 each, and ooze playability out of their ears. Yes, there are more games for PS2 than for XBox, but games are £40 or so each, a HUGE rip-off in my opinion. But then again, so are PC games.
Shop around. Gamestation seem quite open and honest as to the quality (or lack of) of consoles and games, but I trust an independant family-owned games shop more than a chain. The bloke in the local one (A&B Games) actually tried to convince me NOT to buy a certain game once, saying, "It's rubbish, and I'd feel guilty selling it."
The best deals could well be on the web these days. Bit late for Chrimbo, though.
I'm with Fishsta on going for a retro console. I just got myself a Dreamcast for the princely sum of £22 with an extra fiver for the Visual Memory Unit. There are some cruddy games for it but the same can be said for XBOX and PS2. Places like Gamestation and The Computer Exchange normally have some titles in stock for Dreamcast and Saturn (and if you want to go really retro, Mega Drive, SNES, Master System and NES!) but don't bother going to the 'mainstream' stores like GAME as they are unlikely to stock retro stuff.
Still, I'm guessing Despot's younger relation is probably saying PS2 or XBOX. In which case, if he (or she) definitely wants one of those two, you may as well toss a coin. Pros and cons to both.
Having said that, if it's one of the more modern consoles, Gamecubes are now only £80 with a range of good games, BUT they don't play DVD or music CDs (whereas my £22 Dreamcast does!)
At the moment X-box is definetly the way to go. PS-2 is looking aged and Sony have noticed this with a replacement due 3rd quarter 2004. Microsoft of course won't let this go unchallenged so expect an announcement from Redford soon regarding a replacement, they're apparently waiting on Sony finalising the spec for PSX (PS-3??).
Whilst still agreeing with Fishsta in going for a retro console, I did eventually buy an XBOX as well (though at 40 years and 9 months, I may be considered one of the 'older' gaming generation!). I'm not going to say PS2 is rubbish as I've never owned one, but XBOX does for me. Bulky and heavy it may be but you can pick one up 2nd hand and add a DVD attachment for around £100. On games like Project Gotham Racing 1 and 2, (which are basically 'upgrades' of a classic Dreamcast racer, Metropolis Street Racing) Links 2004, and many others no doubt, you can have your own music tracks playing in the background too.
As I said, it does for me, but in the end, you pays your money and you takes your choice, though sports (in particular; football) game fans should note that the previously Sony limited Pro Evolution Soccer series will be coming to XBOX later this year.