At home I have a simple network setup containing 2 machines.
On one machine i have a site hosted with IIS7. Rather than the standard localhost/index.htm
127.0.0.1 always points to localhost. On your home network you should have an IP address assigned by your internet router (dsl/cablemodem/whatever). You need to bind your website to this address. You should then be able to use the machine name to get to the website, but I would recommend actually editing the hosts file of the client computer in question to point a specific name at that computer. The hosts file can be found at c:\windows\system32\drivers\etc\hosts (use notepad) and the entry would look like:
192.168.1.1 mycomputername