gooberking: Specifically what games are in your GOG inventory that you are playing? What game would be a deal breaker if it did not work? What game does your system have the most trouble with, but you put up with it?
farlark: The biggest deal breakers are pretty easy on the hardware.: FTL, fallout 2, planescape: torment, alpha centauri, etc.
The ones that gave me trouble were the relatively newer games: farcry 1 did ok, but it would chug after a bit. Painkiller was unplayably chuggy. Hotline miami ran too slow. Tiny & big and unmechanical wouldn't run at all (probably because there was no graphics card).
Farcry 1 in particular is the answer to your last question. I played it all the time on that poor little laptop. If that game ran smoothly, I'd be pretty happy with my rig.
I'm going to suggest that there is no need for you to buy an actual, physical video card and that a good integrated solution like AMD's APUs will probably more than cut it for what you are looking to do. Look for a system with this chip as a minumum.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113282 And this one as a max
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113288 I would suggest the latter as the CPU is much better even though the video isn't, but it will give you many more options and room to grow. Both should be 1.5-2x as fast as my Intel HD 4000 which I said plays more taxing games than you are playing. The part you want to take to the store is the "AMD APU" and "Radeon 75xx +"
Since you don't really need a video card (unless you do plan on growing into higher end stuff, or just want the best deal on parts) then think about if you really want a desktop at all. I'm not knocking it, but if all you want is to play farcry 1 then you don't need a DT to do it. This spoken from the perspective of what you have, you wanting something modest and affordable. This isn't a gamer's dream rig, but it should smoke a lot of classics. It's possible you could be happy with a little less, but too much less and you would be right back where you started.
I will also mention you aren't going to find many if any prefab systems with only 1-2 gigs of RAM unless its something like a Chromebook, which certainly won't contain any real graphic power. It will have 4-6GB, and probably a 500GB HD. They tend to be more generous on that stuff, giving people more than they need long before they hand out real vid power.
BTW I'm recommending against Intel on the grounds that I think you would end up with the HD 4000 which I have enjoyed for low spec gaming, but the AMD ones should give a 50-90% boost in the graphics department.
Oh and I am being somewhat in contradiction in that non-desktops won't necessarily have my suggestions as options.