@PaulS: mDNS is a service introduced by Apple (as Bonjour) and now recognized by most PC operating systems. In such an environment each devices kind of broadcasts it's name on the network with a defined protocol.
@OP: The only library I know that is able to do mDNS is Arduino Ethernet Bonjour. I never used it so you have to try it to check if it's doing what you need.