Not really much that can be done with 150 US dollars, 150 pounds, maybe...but prices are higher in the UK.
Do you have any existing computers or computer parts that we can use as a base? If we could recycle a hard drive, case, and powersupply it might be possible to build something that can play modern games on low settings.
For $150 you're gonna be a bit hard pressed but I think it can be done. Look wherever you can for freebees...an old tower case you can use, a PSU, etc etc. You could even ask gamers for their old parts on buy/sell/trade forums. I'd gladly give you some old parts out of my machine in the spirit of Christmas but its in storage on the other side of the country and I probably won't be back to it until after the holidays.
Start looking for older CPU/mobo combos on eBay. I'm still running a Q6600 to this day (and plan to for the next few years even). It's a great processor and still relevant. I've seen CPU/mobo/RAM combos go for as little as $70 recently, perhaps less. There's one on there now with bidding under $60. You can give it a mild 3.0 GHz OC no problem and be good to go.
150 GB Raptor drives can be had for pennies on the dollar these days and they're solid, reliable and fast for what they are. $20 or less, shipping included, if you search around.
As far as GPU I have an 8800 GTS 512 g92 in my system and although it's ancient by modern standards it ran everything I could throw at it up through 2012 at 1680x1050. You can get one of them on eBay for about $30 including shipping. If you go this way make sure it's the 512mb/g92 and not the older 640mb/320mb GTS card (which was garbage even in its own time).
For about the same price as a g92 you can snag a Radeon HD 4850 512. It's a better card in nearly every way and it made nVidia sweat for a while until they could come up with something to replace the 8800 series.
Going one step farther the 4870 1GB is about the best card you can hope for within your budget. As far as 2013 games go it's the bare minimum. I'm seeing buy it nows for $49 and you can probably win an auction for even less.
Keep in mind even with these ultra frugal recommendations you'd still only be left with $20 or so after buying CPU/mobo/RAM, HDD and GPU. Any way you hack there's really no way to build a full system unless you're starting with some basic components like case, PSU, monitor, etc. Sure you could probably slap something together with a Pentium 4 and a 4200Ti but don't expect it to run anything outside legacy games like Quake II and similar.
For all practical purposes and for what you're trying to achieve I'd consider the Q6600 and an HD 4850 the absolute minimum, with a 4870 1GB much preferred. It'll run anything up until about Q3 2013 in varying degrees of acceptable quality and gives your son countless awesome titles to keep him busy. But, in the spirit of keeping it real, any $150 box (or similar PC running dated components) just won't be able to take next gen games like CoD Ghosts and similar. Bioshock Infinite on a lowish setting was the swan song for any such systems.
As I said before your best move might be to post on forums (that allow it), classifieds and even Craig's List for free components. Tons of gamers have old mobos, CPUs, RAM, GPUs, etc etc, just sitting around collecting dust because it's not worth the effort to list for the paltry sum they'd get. Tap into the season of giving and offer to pay shipping for any free parts explaining you want to build a rig for your son but have a small budget. List some of the stuff you're considering like a Q6600, 4850, etc to give people an idea of what you're trying to accomplish. I think you might be surprised with the response.
Good luck to you my friend and no matter how it goes enjoy the holidays!