This may be what you've already tried, but just in case ...
To make the phone fully reset itself, you need to hold down the power and home buttons for at least 10 seconds (this cleared an odd little 'phone won't come back on after the battery ran down' issue I had).
Syncing on iTunes shouldn't matter either way, at the end of the reset process you went through, you should have the usb screen on the phone to tell you to start the initialization routine (plug and and let itunes do it's thing).
It's still also possible (if not likely) that the cable could still be bad. The power to charge comes in over different wires than the data that's used to sync / recognize, etc. the phone.
Like I said, you may have already done the full hard reset, if so, I can't think of anything else offhand.