Deltanine

I had an old Pentium II laptop with a broken screen lying around. With 64MB of RAM it wasn't much good for windows, but it does have a 20GB hard disk which makes it useful for a small server such as for websites. So I connected it temporarily to a monitor for the instalation and setup, but now it can run happily on it's own without a monitor.
Follow these instructions if you want to do this yourself:

Follow up:

First, download Ubuntu. I chose Ubuntu because it is excellent at hardware detection and is extremely easy to install additional software on. Burnt it to a CD and boot from it and install it. The installation is far simpler than installing Windows of any flavour.

Once Ubuntu is installed go to the Unofficial Ubuntu Guide and read the General Notes before you get started - especially if you have no experience with Linux. Then follow the instructions for adding extra repositories (the places where you will get the addon programs from).

After that you probably want to complete your LAMP setup - (LAMP = Linux, Apache, MySQL, PHP), everything you need to run a reasonably powerful modern website.

Install MySQL first, then Apache, then PHP, and finally the MySQL hook into Apache. Using apt-get it is extremely simple to download and install software - I've installed these programs on a Windows system before and it was way more of a hassle.

If you have followed these instructions you should now have a functioning webserver.

Additionally I wanted to be able to upload and administer the website remotely by via FTP, so follow these instructions to install FTP, and then follow these instructions* so you can upload and download from the directory with the website in it (probably /var/www) - be sure to put "/var/www" instead of "~".

Now a laptop with only 64MB RAM will function even with Gnome desktop running, but it'll run faster without it.
To stop Gnome running normally you'd press ctrl-alt-backspace, however that jut restarts Gnome as it's in the startup scripts. To prevent it restarting install rcconf (apt-get install rcconf) then run it from a root terminal and use use the space-bar that to remove the star next to the GDM startup script. You will now be able to close Gnome without having it restart. Note - if you want to start Gnome just type in "startx".

All you need to do now is register with dyndns.com and set up your computer to send them your IP address automatically. Some routers have this feature built in. If your computer is behind a router you will also need to forward port 80 to your webserver's internal IP address (it helps to make it a static IP on your LAN).