You can run it on almost anything you like. The resources requirements of a LFS dedicated server are tiny. You could happily run it on a 400MHz machine running Linux/*BSD/OpenSolaris (headless), or windows 2000.
The real issue is bandwidth. If you don't have much or it's unreliable, then I wouldn't recommend running a server from home. You can use the calculator in the non dedicated version of LFS (the normal version) to give you an idea of how much you need (Multiplayer > Start New Game, configure your parameters and you'll get an idea on the left). There are also providers, such as 500servers, who will make things much cheaper for you in the long run (especially if you only plan to host a LFS server, nothing more).
I also wouldn't recommend using a laptop as a server - they just aren't really suitable or designed for continuous running, especially in confined spaces.
If you want to build a small box just for LFS, then I can happily recommend almost any of the VIA Epia mini-itx or smaller system on a board solutions. I've run a full server from one (an 800MHz box, with a single IDE hdd, 512 MB of RAM) in the recent past and it used practically no resources. With LFS, currently, the real hog is bandwidth.
Hope that helps.