What does "Local Domain Name" on router do and how do I get it to work?

Solution 1:

"Local Domain Name" refers to the domain name that the DNS resolver on the client will try in order to find a DNS entry; it does not create a DNS entry on the DNS server. If you have entries in an existing DNS server for "foo.lan" and "bar.lan" then a Local Domain Name of "lan" would let you resolve them using "foo" and "bar" respectively.

Solution 2:

The "local domain name" is used for the router to provide normal DNS service. For example, on mine it's set to kf8nh.com so machines on my network are all identifiable as foo.kf8nh.com where foo comes from the DHCP client ID or static DHCP reservations. If you set (as DHCP normally will) the router as your DNS server, hostnames like that should work; if you also then set the domain name or search path on your computers to the same value then you can use those names without the domain. (I don't know why many/most(?) routers don't send that with the other DHCP options.)