Don't bother is my general advice.
If he's playing on an HD telly he won't be able to play in HD when recording on a low end cappy, the cheapest solution would be a Blitzbox HD which doesn't record in HD but allows a component signal through, at about £100.
If he's playing in SD, or doesn't mind playing in SD (get him to try it first, it's deceptively irritating) the easiest solution is a Dazzle Platinum, with the right render settings it can be decent quality and isn't too much.
If it's because he doesn't mind SD, then beware that over time he may want to upgrade to an HD card to be able to play and record in HD, as he'll most likely want to start uploading to youtube, and without HD or a big shoutout from a big community member (i'm assuming this is CoD) it will be very hard for him to get subscribers, and at 12 nobody's going to be giving him recommendations.
HD PVR's and Blackmagic's are the youtube standard at the moment and if he's interested in getting into youtube I would try and talk him out of it, the youtube community is so saturated right now a twelve year old will find it very hard to get noticed, and i'd be surprised if he is particularly good, good scores don't mean somebody's good at the game.
The Blitzbox HD would be fine, but if he's using it for youtube he'll want to get an HD Capture card later on to try and get some more subs, and if you go with that make sure he's totally in agreement that he's not upgrading to a "better" one later on, as before long the HD PVR's and Blackmagic's will be replaced by 1080p capture cards anyway.
If he's just recording short clips for facebook uploads etc. (no idea why you'd put xbox clips on facebook) then a dazzle will be fine, but dazzle's usually either end up being novelty items and getting boring, or gateways to inspire an immense desire to get an HD PVR, very few people are constantly happy with their Dazzle..
