A discussion at: http://www.reddit.com/r/todayilearned/comments/37fzb6/til_that_in_2013_a_kenyan_lawyer_filed_a_petition/
made me wonder about this.
I believe that the only person left with any tie to the Roman government is the Pope.
The Vatican, Germany, Turkey, Greece and Liechtenstein would have the best claims I believe. The Pope should be obvious. Germany is the HRE's successor state. Liechtenstein was a HRE membeg state. Turkey and Greece are successors of the Ottomans, in turn of the Sultanate Rum and thus Byzantium.