Both systems are matured, with good title selections. The X-Box is slightly newer, and with better specs, but the difference is mainly the hard disk and the availability of mods/hacks. Microsoft is quite ardent on getting exclusive titles for the X-Box, so it will see a few titles that you won't see on the PS2, but that goes both ways; developers are not keen on loosing out on the larger PS2 market. If you - sorry, your kid - like japanese games - Square, Namco, etc - those titles are more likely to be PS2 exclusive.
The gamecube, which is the newest of the three big names, might also enter the equation but I know little about it except that the pricing is friendly.
As far as the cost of the two is concerned, at least looking at EBGames, cost is roughly the same for both consoles, considering both the best sellers and console systems themselves. The X-Box often comes bundled with the DVD set, but the Silver PS2 also comes with a remote controller.
Both are easily made DVD-zone selectable... a plus for me but probably less so for you.
In early 2004, Sony will launch an interim version called PSX - it's still a PS2, but with hard drive, improved processors, digital image editing, surround system, internet, and so on. It's somewhat expensive, pricing is initially at 719 dollars, but is labeled a 'digital appliance' - probably not very useful if you have most of the functionality allready. I think that's more of a footnote; you could buy a PS2, dvd player, AND pc for the same amount... not in such a neat, living-room-friendly package, but still.
The PS3 is scheduled for launch in 2005.
The XBox 2 is on the rumor level still; nothing so reliable I really want to post it but I've heard 2005 and 2006 mentioned.
As for unit sales, I do not know the exact figures; the PS2 reached 50 million christmas last year, while the X-Box reached 10 million august this year.
Right now, which you should buy is a tossup because both systems has at least two years left in them, but allready next year it'd be time to consider waiting for the next generation consoles.
As far as the kid is concerned, he'll likely be happy with either. If he gets a PS2 he gets access to a large amount of games from his friends right away, while with an X-Box he'll have games his friends doesn't... something that might be a plus.
An aside on the internet stuff:
I have a PS2; the X-box live functionality is of marginal interest since I have a PC for that, but supposedly, the X-Box is superior in this regard, or at least got off to an earlier, and smoother, start, but supposedly the PS2 is definitely playable on the net as well.. SOCOM was the big name for that on the PS2, where you could paly with a headset and issue voice commands to both teammates over the net and AI soldiers (through voice recognition). Which is sort of neat, but I never tried the damnthing.
On the other side of the coin, games launched both for the X-Box and PC, like MtG: Battlegrounds, can be played platform-independently; you can, on your PC, play against someone on an X-Box.