I think for most game-testing jobs, you have to have a graphics design degree (Or something else video game related) because they make you write about any glitches you see and provide suggestions on how to fix them.
You've got the right of it... True, you can sometimes find beta-testing slots where you get to play the game for free, but they usually don't pay you, esp. for any beta-testing in public. The reason is that there is such a high demand to do exactly what the OP wants to do... get paid for playing games. In fact a lot of the major gaming publishers will use different conventions or promotional activities for their public beta-testing; events like E3C, Comicon, or even the company's own publicity event will draw a large pool of potential customers, allows them to play for a bit to see what it is like, and then they can get the opinions from these public players to use in a beta-testing analysis. Some companies may use survey sites, but I'm not sure if they would actually send you a game to try, or whether it would be more along the lines of watching trailers and / or other advertisements for the game to see if it would appeal to their target audience.
If you actually work for the company doing alpha or beta testing, then you need some type of computer degree, and sometimes a degree or business certificate in qualitative analysis or quality testing. The true game testers make it into a science, mapping out a strategy to test out several various aspects of the game within a single session. Some of the testers (esp. alpha-testers, or component testers) don't really "play" the game as much as they try to break it; they may try walking thru walls, killing a major protagonist, buying a negative amount of equipment (to make money), etc. And a key part to it all is reporting on the findings from the quality tests. The reason for the testing is to be able to go back to the design team and have them fix bugs that were found, make tweaks to the gameplay, improve on the storyline, etc. Depending on the age of the kid, the company may not want to use them for testing just because the kid hasn't developed enough writing or communication skills yet.
The only other type of games that you can get paid for are either for advertising sites or for gambling sites. Advertising sites, like Winster or the PTC Game page on FC, get money from advertisers to play their ads while the people play games, or by people spending money on the site itself or thru linking to the advertising site and making a purchase there. The gaming site then turns around and pays their players a certain percentage of what they bring in thru advertising dollars, either by awarding prizes or by actually paying a fixed amount. And of course with gambling sites, you usually have to pay to play and you may win, just like any casino game.
If the OP's son thinks that they may like to do this as a career, then I'd suggest he continue to get involved with technology and gaming. Become an active member of the gaming publisher's forum; this not only helps to improve his writing skills, but it may also send a flag to the company that this person has a passion for gaming. Have him try to submit suggestions or bugs on these forums, if they have a place for it, esp. if not only he reports a bug but also a possible solution. Check out the company's jobs listing for public beta testing, and sign up for it. Even if it isn't a paid position, it again shows the company that your son want to be involved with game development, and often the company will have some type of credits listing all of their beta testers that your son can use when interviewing other companies.
He may also want to get involved with game "modding"... "Modding" (the gaming lingo for "modifying") is where the player makes tweaks, develops new levels or quests, or even redesigns the graphics or music of a game. Some of the newer games coming out (esp. for the PC) will even include a development package to allow players to make game mods, and may even have a site to host these mods that the different players have made. Sometimes the company will even award prizes for the best mod (esp. if they run a contest centered around a specific aspect of the game or a particular theme like a seasonal holiday), or may even turn around and make it into an official extension for the game, giving the modder royalties. Being in the modding community will again show potential employers that your son's passion for gaming has evolved into wanting to make games better via testing and/or development.