You would need a catastrophic event to happen to the Empire's central government during a period where they still have firm control of Southern Italy. The 4th crusade constituted just such an attack on the central government, and allowed four states (Epirus, Nicea, Trebizond, and Theodoro) to form from the contenders attempting to take the throne, two of which (Trebizond and Theodoro) were never really reunited with the empire proper, but outlasted it anyways. If you had this event (most likely a sack of Constantinople) take place while they still control Naples, and the people who attacked Constantinople are for whatever reason unable to take southern Italy as well, then it will almost certainly give rise to some ambitious noble who sets up an empire in exile there. He and his descendants will presumably now have the strongest claim to being the heirs of Rome, provided that they outlast any other successor states, and that none of those successor states retakes Constantinople. After that it is just a matter of having an independant southern Italy into modern times, which is certainly possible, but a state like that is unlikely to be very powerful, so it would be the Roman Empire in name only.