Multi-client to server connection demonstration
If your students are old enough to own phones and send text messages, then perhaps you could use group-texting as an example?  One person in the group sends a message to the others; the message goes off to the company's server, which broadcasts the message to the people in the group?  Depending on their age, that might be an example to which they could relate and that would engage them...

I guess it depends on how deeply you want to go.  You could delve into the details of broadcasting vs multicasting, in which the sender of the message (the server in your example) sets the correct bits in each packet and the underlying network takes care of ensuring that the right packets get to the right clients.  

If there is no network-level support for broadcasting or multicasting, a "broadcast" among N clients can still be done fairly efficiently in a distributed manner, without involving a server. For example, suppose N is 8, the clients are numbered 0-7, and 0 (which could be your server) wants to broadcast a message.  Then...


At time 1: 0 sends the message to 4 
At time 2: 0 sends the message to 2 and 4 sends it to 6 
At time 3: 0 sends the message to 1, 2 sends it to 3, 4 sends it to 5, and 6 sends it to 7


This approach can "broadcast" a message among N clients in O(lg(n)) steps, so long as the network allows different clients to send simultaneously.  What's happening there is quite different from your cups-and-string demo...