MrSpock -- as long as it can be traced back to Ivan, whether that be through a tropical wave (like Bonnie's regeneration), mid-level vortex & vorticity maximum, or some other means of continuity between the two systems, I believe the NHC would rename the system Ivan. Even if a storm goes extratropical and then becomes tropical again (very rare), they keep the name. Only if they change basins without maintaining themselves at least as depressions or if the main system completely dies and a new system forms that might be related to the old one (but evidence is lacking) would the storm get a new name.

That said...I don't get the feeling Ivan is going back towards the Gulf, thankfully.