First thing: Possible Hardware or Software Conflicts with iSight.
Chaining external iSight with other Firewire devices on the same Firewire ports often causes function problems. Disconnect all other Firewire devices and restart your Mac while testing your iSight with your mini. (I think your mini has only one FW bus. If that is correct, you may need to disconnect all other FW devices every time you want to use use an external iSight with your mini.)
If your trouble continues when no other FW device are connected, consider these other possibilities:
(1) If the old external iSight you have still works, it should work with the new Mac mini. You can test an external iSight using the suggestions in this link , including Apple's troubleshooting suggestions from http://support.apple.com/kb/HT2090
Test more than one Apple camera-enabled app in more than one user account. However, don't use the "Guest" account for testing. If your Mac only has one account, create a new user account for testing.
(2) Although unlikely, it is possible that your Firewire 400-800 cable is not working properly. You can check cable function by trying the cable with a different FW400 device.
If you don't have another FW400 device, get at least one more connector to test for possible connection problems. Any working one of something like those listed here should work to connect your FW400 external iSight:
• This Firewire 400-800 cable can work:
http://store.apple.com/us/product/HA352ZM/A/ (I use this one.)
• Some users report that this low-priced alternative cable works:
http://www.newegg.com/Product/Product.aspx?Item=N82E16812189097
• To use your iSight's FW400 6pin-6pin cable, you can use a 6pin-9pin adapter like one of these:
http://tinyurl.com/ydsbb6m (I successfully use this one, too.)
(3) Apple discontinued external iSight several years ago.
My external iSights (I have two) are great webcams, but they are getting old (as consumer electronics go.) Both of mine still work as well as they did when they were new. However, some previously owned or even NOS iSights may have been damaged by rough treatment or improper storage.
The only way to know whether any presently available external iSight works properly is to test it using suggestions similar to those I offered above before you buy. If you did not test the external iSight before you bought and now find it will not work properly with your system, perhaps a trusted seller will give you return or exchange privileges.
Otherwise, if you don't want to spend the money to have your external iSight serviced, consider a new, Mac-compatible USB cam that offers a current manufacturer's warranty instead of any external iSight. Some examples of current compatible cams are described at http://www.mac-compatible-web-cam.com/
(4) If you want to look for more ideas, you can search iSight Discussions.
http://discussions.apple.com/thread/3983993 is a good place to start.
(BTW, current FaceTime software versions work with external iSight's mic.)
(5) If you need other suggestions after doing your own searches, we will try to offer more specifics based on the details you supply in your reply here.
Message was edited by: EZ Jim
Mac OSX 10.8.2