For your budget, a Windows 7 Home Premium 64 bit System Builder (generic OEM) copy may be the best choice.
Such a copy is tied to the first system on which it's installed. The system is basically defined by the motherboard. See here: Licensing FAQ
"If the motherboard is replaced because it is defective, you do not
need to acquire a new operating system license for the PC as long as the replacement motherboard is the same make/model or the same manufacturer's replacement/equivalent, as defined by the manufacturer's warranty."
From online reports, Microsoft is fairly liberal about this. They have activated OEM copies on repaired PCs, where the "repair" was a 100% hardware replacement.
A minor consideration is that a system builder copy only includes one bit depth (32 or 64). I have read claims that a license key can be used with either, but the box only contains one DVD.
There's one oddity of the current "system builder" license. Supposedly, it is for use only on systems intended for resale: Is it OK to use OEM Windows on your own PC? Don't ask Microsoft | ZDNet
Putting a system builder copy on a system for your own use is apparently a violation of the license agreement. As far as I know, there is no enforcement of the resale requirement. I recently purchased an OEM copy from a major US online seller, and they didn't even bother requiring a token hardware item to be purchased with the copy. I think it must still be regarded as a gray market copy. (Grey, in the UK.)
It's also possible to install and activate
an upgrade copy on a new system without having a qualifying OS installed. That would be an explicit breaking of the license agreement, but you'll find a sticky in this forum that tells you all that you need to know about how to do it. At least one method (double install) requires no hacking. It's easy, and I doubt that the result would be distinguishable from a legitimately installed upgrade copy, but I still regard it as piracy.