If a computer's host name is "giggidy", then when it requests a DHCP lease, it sends that name to the DHCP server. If your DHCP server is hooked up to a mini DNS server, then the DNS server will resolve "giggidy" or "giggidy.whatever" to the IP it was assigned. This is how dnsmasq works, and a lot of consumer internet routers, too.

The router's IP and a nameserver IP are the only ones you should have to remember in the event of problems, and both of those can be short and simple.