Apple Mouse won't work with Mac Mini

I have a third-party keyboard/mouse that I use alternately with my 3-month old Mac Mini and a PC, but I have an intermittent problem on the Mac Mini. Often the mouse keeps becoming unresponsive. Sometimes restarting the Mac fixes the issue, but only temporarily. It always works fine on the PC.

To to resolve this problem, I tried using a Apple Wireless mouse with the Mac. I took the batteries out of the third-party mouse, put fresh batteries in the apple mouse and turned it on, and restarted the Mac. When the Mac booted up, it still would not recognize the Apple mouse. I had to put the batteries back in the third-party mouse to do anything.

Next, I opened up Mouse Preferences using the third-party mouse and clicked "Set up Bluetooth Mouse." The gear icon spins and spins, but it keeps saying no mouse found. Finally, I tested my Apple mouse out with my older iMac and it worked perfectly.

How can I resolve this issue? Thanks.