Yes, any version of Windows 7 (other than Starter) would probably do. If you purchase an upgrade version you might need to install it twice to be able to activate it (I'm not sure if the installed Windows 8 would qualify for a Win7 upgrade) --- How to Clean Install Windows Vista with Upgrade Media | Windows Vista content from Paul Thurrott's SuperSite for Windows
--- A "System Builder" version will work on any system.
The OEM versions that sobeit mentions are those that come pre-installed on a major-brand computer (Dell, Sony, Toshiba, Gateway, Acer, Hp, Lenovo, Asus, Samsung, etc.). Those licenses are tied to the hardware of the computer they come with, and can't be transferred to another PC (within the terms of the license).
money is tight at all, you can always have a go with Classic Shell or Start8 first, just to see if that makes the new computer fun again. The "guts" of Windows 8 have a fair number of performance and security improvements that are noticeable. It's unfortunate that the 'Metro' interface wasn't offered simply as an option that could be turned on or off. From the desktop, Windows 8 and Windows 7 behave very much alike. It's the functionality of the Start Menu that is the main difference (and which can be overcome by either a menu program, or user "tweaks").
Best of luck
. . . Gary