Well, according to Wikipedia, Constantine XI had a nephew named Andreas Palaiologos.
http://en.wikipedia.org/wiki/Andreas_Palaiologos
His father Thomas continued to rule Morea until it fell to the Ottomans in 1460. Since Constantine never named a successor, I think the throne would go down dynastic lines, but that was never part of Roman law. They just went along with it because it made things easier with the army to cultivate a dynastic sentiment. So assuming the title didn't pass to Mehmet when he conquered the Empire and began calling himself Kaisar-I-Rum, Andreas would be the rightful Roman Emperor, and that whoever he named as his successor is his legitimate heir. But according to Byzantium - The Decline and Fall, by John Julius Norwich, Andreas sold the title to Ferdinand II of Aragon and Isabella I of Castile. So the House of Trastamara in Spain might be the heirs to the title, but their house ended in 1501. The Spanish throne would then pass through marriage to the House of Habsburg.
Donald M. Nicol's The Immortal Emperor mentions a Constantine Palaiologos in the Papal Guard, and lady named Maria, who wed a Russian Noble named Mihail Vasilivich as possible offspring, but going by the principles of nominative succession, they aren't included as candidates for the crown. After the Trastamara Dynasty ended, the Hapsburgs ruled Spain, but whether the title applies to a Habsburg ruler of Spain or the house of Habsburg in general is a whole different can of worms. Assuming it's simply Habsburg kings of Spain, then the title as of the late 17th and early 18th centuries was this sexy devil:
http://en.wikipedia.org/wiki/Charles_II_of_Spain
Charles II died with no successor, and the line of Spanish Habsburgs died with him. He had named a great-nephew, Philip de Bourbon, Duke of Anjou, as his successor, the grandson of Louis XIV. The other nations of Europe didn't like the idea of Spain being effectively ruled by the King of France, and this touched off the war of the Spanish Succession. Eventually Phillip would be allowed to keep his throne, and is known to history as Philip V, King of Spain. So assuming the title passed to the Bourbon rulers of Spain, that would make the current Basileos King Juan Carlos I of Spain.
Assuming the title of Byzantine Emperor was the property of House Habsburg as a whole, then the last Habsburg was Maria Theresa of Austria. With her death in 1780, the house of Habsburg was seen as defunct, and was replaced by the Vaudemont branch of the House of Lorraine, which would style itself Habsburg-Lorraine. At the time, the head of House Habsburg-Lorraine was Joseph II.
http://en.wikipedia.org/wiki/Joseph_II, ... an_Emperor
So if the title was his by rights, then Eastern and Western Empires were again united in a single person. The Holy Roman Empire, and the Title of Holy Roman Emperor, would be dissolved by this man.
http://en.wikipedia.org/wiki/Francis_II ... man_Empire
Assuming the title was now one with the office of Holy Roman Emperor, Francis II was the last Byzantine Emperor. In 1806, Francis II abandoned the title of Holy Roman Emperor. All the subsequent titles, mainly Emperor of Austria, had nothing to do with Byzantium, but if we include the title as legacy baggage, then the last Byzantine Emperor was Charles I of Austria, after whom the Austrian monarchy was dissolved.
The Habsburgs themselves would go on until November 11th, 1918, where Charles I of Austria-Hungary renounced participation in state affairs. While not an official abdication, the house of Habsburg-Lorraine would be officially dethroned by Austiran law in 1919, and Hungarian law in 1921.
So we have a number of potential successors. The now-defunct house of Trastamara, the now defunct House of Hapsburg-Lorraine, the House of Bourbon ruling modern day Spain, and this doesn't begin to calculate Napoleon's reorganization of Germany which dissolved the Holy Roman Empire. It gives us a number of disparate dates for the end of the Byzantine Empire as well, be it 1453, 1461, 1501, 1780, 1806, 1918, 1919, 1921, or continuing in rump form to the present day.
All this assumes that my wikipedia info is accurate, and that I didn't make a mistake somewhere. If anyone finds a mistake I made, please point it out.