It's not that common for an iPod shuffle to work on one Mac, but not another. Usually, if it does not connect with one Mac, it has become faulty and does not work period.
Have you tried connecting something else to your Mac's USB ports, to make sure they are working properly. Something like a USB flash drive would be ideal for the test?
Since it DOES work on your wife's Mac, the shuffle seems to be working, and the cable being used is good. If you run System Profiler (or System Information with Lion) from the About This Mac window, or from the Application/Utiltiies folder, is the shuffle seen as an "iPod" under USB?
Maybe you should do a Restore on the shuffle (using iTunes) while it is connected to your wife's Mac. Doing so erases the shuffle, re-installs its software, and sets it to default settings; it will also sync it to that Mac's iTunes library. Then, connect it back to your Mac to see if iTunes now sees it. If it does, you can sync it back to your Mac.
If it does not work after a Restore, then the problem is likely to be some type of software issue on your Mac.