Well, it's a little difficult to say without knowing what it is your brother is trying to do with the system.
At an old job, there was one guy who was always complaining about how slow his system was. They had a 4 year rotation cycle for getting new computers, and this guy was probably only a few months from a new one. Anyway, he absolutely insisted on having things like Adobe Creative Suite... The whole thing... Open all the time, and a bunch of other large programs. All on about 2GB of RAM I think. No less than about 4 different people went out there to look at the system, kept turning off a bunch of things the guy had running, and imagine that, the system ran much better. Until he turned around and started all of those things back up after the tech left.
Then there's the fact that eMachines is now an Acer brand, and Acer isn't exactly known for building quality systems. They've been fighting it out with HP for who can build the worst system ever since HP and Compaq ended their battle in a merger.
Something I always tell people, is that with electronics of pretty much any kind, you get what you pay for. You buy the lowest of the low end, and it's going to be a pile of rat droppings. Any computer that has an MSRP of under $500, you don't want to touch with a stolen 10ft pole, because that is a loss leader system. They are selling it for less than it costs to make it, and the idea is they can make it up on things like accessories and extended warranties. But if you know you're going to lose money on every unit sold, how much would you invest in making sure it was a solid bit of hardware? It would probably end at whether or not it powers on.
So, while you can't rule out malware or things of that nature, there's also a good chance that your brother basically bought himself a headache in a box. With electronics, you can either pay more up front in cash, or you can pay in installments of torment and frustration over the course of owning the device.
And I'm going to have to disagree with everything speedy said. A SSD is not likely going to help you much, and there's really no point sinking much money into that system. And virtually any performance boosting program you could find at a Walmart will be all but certain to make things even worse.
Finally, defragmenting has little to no benefit. It's one of those bits of information that stopped being useful to all but a select few people many years ago, but simply refuses to die. Windows is the only OS that still bothers to even include such a tool, and even Microsoft has gone to pretty great lengths to try and bury it deep in the scary technical areas of the OS where most people won't dare tread.
If the system is locking up, start by updating the drivers. If that doesn't work, then start cutting back on high end programs like games, see if things stabilize. Make sure you open the case and clean out the dust bunnies with a can of compressed air. That's something you should do roughly every 6 months, so odds are your brother's system is long overdue. But he should be bracing himself for the possibility that he bought a low end POS, and this is as good as things are ever going to get with that system. Live and learn. Next time, you don't go shopping for a computer based solely on price... And if some system seems to have really good specs at a really low price, run away as fast as you can.
Next time, I'd suggest avoiding any sub-$500 system, regardless of brand, unless it's a short term promotional price. I'd also avoid any HP or Acer brand, which includes Compaq, Gateway, and eMachines along with HP and Acer obviously. If that means you have to save up a little longer before buying a system, it'll be worth it in the long run.