“These are the voyages of the starship Enterprise, it's continuing mission to explore a strange new worlds, to seek out new life and new civilizations, to boldly go where no one has gone before.”-Gene Roddenberry
I suggest looking for something that uses an AMD APU processor (A6 or A8 for example). From my experience, Intel HD 3000 graphics is the absolute minimum to play minecraft at far render distance at 1080p. However, that isn't available in many desktops (almost any desktop you see that advertises Intel HD graphics uses the 2000 model which is even more underpowered). Thus, I suggest disregarding Intel altogether for your budget and looking at AMD APUs (not their athlon or sempron line though).
Also, if you are looking for a laptop, same advice
And, if you do feel up to building your own desktop, people here will be more than happy to "help" you.
Operating system, 100 bucks. i3 2120, 124 bucks, case 40 bucks, motherboard 60 bucks, ram 20-30 bucks, HDD 60 bucks, Optical drive 17 bucks and that is just est and not gonna happen.
A AMD APU laptop is best choice you can go. Still searching for a good brand and deal on one.
lol. Yep. That does add up to $431, although the numbers you listed are obviously best-case scenario prices, considering tax, shipping, stupid SATA cables, etc, it of course gets more expensive. But, the OP could potentially do that if he wanted.
As a note, for only a couple dollars more you can get the i3-2125. It is the same as the 2120, but with Intel HD 3000 graphics; worth the upgrade in my opinion.
MS Windows 7 Ultimate 64-bit | AMD Athlon II X4 635 | 4.00 GB Dual-Channel DDR3 | BIOSTAR Group A880GU3 | ASUS VH238 | 1024MB GeForce GTX 550 Ti (Gigabyte) | 313GB Western Digital WDC WD3200AAJB-00J3A0 | TSSTcorp CDW/DVD TS-H492A | Realtek High Definition Audio