MrHoffman is 100% correct. Disk Utility repair permissions, verify (or even repair) have absolutely nothing to do with the prompt you are getting.
And reinstalling Mac OS X will change nothing, as the problem, as MrHoffman explained, has nothing to do with your Mac. It is your router and/or your ISP.
The bash shell always does a reverse DNS lookup on your Mac's IP address to get your computer name. The Mac's DNS lookup service does not get a result from the DNS servers, it will use your computer name as specified in System Preferences -> Sharing. In your case, prior to restarting your cable modem, the DNS lookup did not return anything, so your sharing computer name was used. After the cable modem restart, the DNS lookup started returning a result so bash used that information.
Chances are it is a lot more work than it is worth trying to get the DNS lookup to stop returning a value, so it is easier to just change your PS1 shell variable and get on with life.
You could even use the following to always get your sharing computer name
PS1='$(networksetup -getcomputername):\W \u\$ '