I'd say pretty much any laptop will do for developing PHP/MySQL applications on Windows, provided that the laptop specs are good enough to 1) running Windows itself, and 2) running the IDE you want to use
MySQL is pretty lightweight and Apache/IIS is the same. Obviously the amount of traffic matters, but since you're hosting it at home, I suspect the traffic is rather light as well?
Often critical: - Powermanagement (acpi/ apm) - enhenced grafic features (3d) - internal modems - seldom: sound
You wan't to run linux on windows (look at cygwin), or alterating? Both will not depend on your hardware.
What to look at? If I would set up a server at home, my bottleneck would be the upspeed to the internet. The second problem would be, how to find out, what is most important: Speed, security, 24*7 uptime, backups, marketing, ...