is it possible to set up an Apache server on a machine which is part of a LAN, then allow machines on the LAN to access the server via a custom ServerName (instead of Localhost). I want to serve a simple website in an office space using a short ServerName (no ports etc if possible), but I want to make sure this is possible (after originally being certain it was!). THanks in advance.
Yes, this is relatively straightforward.
Most SOHO network routers will allow you to set up a static DHCP record that ensures the IP address of the Apache server will stay the same, or will be quite happy if you assign the server an IP manually, as long as it is outside the DHCP range you're using.
Then you need to assign the name you've given the server to that IP address in DNS. Linksys/ Cisco Business Series routers and some others also allow you to set up local DNS entries that are checked first, before passing the query to the network, or you can set up dnsmasq on the same box as he Apache server to provide a local DNS server.
If you tell us what router model you have, and what operating systems you use, members of the community will be able to offer you more specific advice.