Is there a way to rename a site?

That did not work either @lukptr. I tried adding "site_name" to site_config.json and then doing the bench setup nginx command. What I am noticing is these settings in the config/nginx.conf file

 location /socket.io {
                proxy_http_version 1.1;
                proxy_set_header Upgrade $http_upgrade;
                proxy_set_header Connection "upgrade";
                proxy_set_header X-Frappe-Site-Name site1.local;
                proxy_set_header Origin $scheme://$http_host;
                proxy_set_header Host $host;
                proxy_pass http://frappe-bench-socketio-server;
        }
 location @webserver {
                proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
                proxy_set_header X-Forwarded-Proto $scheme;
                proxy_set_header X-Frappe-Site-Name site1.local;
                proxy_set_header Host $host;
                proxy_set_header X-Use-X-Accel-Redirect True;
                proxy_read_timeout 120;
                proxy_redirect off;
                proxy_pass  http://frappe-bench-frappe;
        }

Notice the proxy header of X-Frappe-Site-Name in there is not changed. I tried to change it by hand, reload and restart and it somehow puts the site into maintenance mode. Probably because frappe does not see the header it expects from the http server in the header. I was hoping the addition of the same parameter in the site_config.json file would do the trick.