I'm not much of an expert on the client/server thing, I get confused too!
Your upstairs computer can remain as it is. What I believe it is doing is catloguing and storing your media collection as a local library and then making it available to other installations of MC. Each client also has its own local library but you can instead load a library from another server. But that client can still act as a server for control purposes. Try enabling Media network on the Mini and it should then be available to JRemote to connect to and control but it will still be using a remote library on another server. The terms client and server for J River purposes relate to the actual function you're performing and this will change according to what you're doing.
If the shared drive is available to the Mini under the same name as the upstairs computer then this is what enables it to play the media using a remote library. If you want to convert the Mini to hold the library instead then you will have to import the media again into the Mini's local library, or backup the library on the upstairs machine, copy the backup to the Mini and restore it there, making sure all the file locations you set up initially are replicated on the Mini.
However, you will then need to do all importing, ripping etc on the Mini which, if connected to the living room TV and a wireless keyboard, might be cumbersome to do. You could connect the upstairs PC as a client to it for management purposes but clients don't have full admin capability to a remote library, and you'd need to set up authentication and synching, which I've never managed to get working properly. (I think JRiver doesn't actually *connect* to a remote library as a client, it just loads a remote library locally, so any changes you make have to be synched back to the "server").
As I said, I'm not that certain of how it all works - you might get some better advice posting in a general thread in the Media Network forum rather than the specific JRemote thread in Third Party. The same considerations would apply if using the JRiver-supplied Gizmo as a remote control.