If you just want to do this for convenience, the simplest method is to add a stanza to your ssh configuration file `~/.ssh/config` :

 Host my_server_name
 HostName some_ip_address
 User my_name
 ... any other options

Then you can `ssh my_server_name` to connect.

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Other options include:

* using mDNS and connecting to the `.local` name your machine advertises (mDNS is called `bonjour` in the Apple world; a common linux implementation is `avahi`).

* editing the `/etc/hosts` file on each client machine to provide a mapping from the IP address to the chosen server name.

* installing and configuring a DNS server on a machine in your local network, and setting it as the preferred DNS server for all other local machines.