For LAN usage, I do not think urbackup server cares at all about server name. Your clients would just need to be able to find the server and the key would need to match. That could be difficult in some situations, but on a basic LAN setup it will just keep working.
For internet clients, I think all you need to do is change “Settings-> General → Internet/Active Clients: Server URL clients connect to:” in the web config of the server. Just have both the old and new DNS names work until all of the clients have updated their local config file. If that can not happen you can update the clients manually: