HP Pavilion H8-1040: Not bad, but that price tag is outrageous for a computer that has a pitiful 5400 RPM hard drive. For that much money they should be giving you an SSD with a 1TB SATA HDD. I would not recommend it. Not to mention the hardware is all outdated: USB 2.0 (3.0 is now available on most MBs), PCIe x16 1.0 (2.0 is the standard), 5400 RPM HDD (7200 is standard and SSDs are the fastest for seek time), plus all the components are probably very generic. Final vote: NO.

CyberPower PC: My friend bought one of these brands, but about a year later I transferred all of his components to a new case that lets the system run much cooler. It doesn't look bad and the components don't look cheap, but the motherboard is a mini, so you'll be very limited when it comes up to future upgrades. The CPU is an AMD Phenom quad core, but the AMD hexacores are extremely cheap, so it's odd that they went with that one. The limiting factor here is the size. Between the small case and limited upgrade room I'd worry about future modification potential and heating issues. Since I like to upgrade video cards and the like, I tend to not like limiting myself with the case or motherboard size. However, if you never do intend on upgrading, this seems pretty decent. Final vote: MAYBE.

Please read this, I'm not sure if it's current but it'll help you select a card that matches the graphics tier you care for:

http://www.ffxivcore.com/wiki/Graphics_Card_Matrix

Don't let anyone tell you a video card (especially anything above the GeForce 250 series) won't run the game. If they do, it's because they want to run everything at 100%, even if it doesn't make a big difference visually. The cost, however, is 2-300 bucks more for that marginal gain. Besides, if they aren't using that card, they really don't know what it can run. Do keep in mind that minimum specs list a GeForce 9600 series card, which is about as crappy a card as anyone might try to sell you anyway.