I personally use avahi (mDNS) as many other replies have suggested.

I use NixOS, so it was easy to make a function to abstract over the config. In each computer's config, all I do is specify a hostname. This function does the work (or really, some nixpkgs committer did):

    { hostName }:

    {
      services.avahi = {
        enable = true;
        nssmdns = true; # Allows programs like ssh to resolve .local domains via avahi
        inherit hostName;
        openFirewall = true;
        publish = {
          enable = true;
          addresses = true;
          workstation = true;
        };
      };
    }