If you buy it, before you get it, right there in the store get them to open it up so you can look over the capacitors on the motherboard. If you see any that are bulging or leaking, it's getting ready to die. Google "bad capacitors" for some pics of them. A few years ago, where I worked we had to regularly replace one Dell machine here, another there, as the capacitors on their cheap motherboards failed.
Hmmmm. I have only built one computer before and that was for myself. I'm not sure how comfortable building one for someone else. I would really like to get a pre-built PC but all the decent new ones come with Windows 8 and i don't even want to go down that path. I told her that She could probably get one for around $400 (just off the top of my head) but i would like to find one for under $350. I'm just stressin about playing around with her money.
Based on a request I made some time ago, c.hegge (who is an Australian reviewer who does competent technical reviews of PSUs) took a brief look at the 300W TFX PSU in the IN WIN slim mini-ITX case. He hasn't formally reviewed it, but based on the components used believes it isn't junk. Accordingly, that case with the included PSU would shave a bit off the cost, and make for an easy little system for Grandma to move around.
I otherwise like G-unit's slightly stronger build though; the very bottom-of-the-pile APU has seemed very slow in every benchmark I've seen.