Well, this is a very complex matter.
Firstly, regarding the royal dynasties:
HRE: The second Holy Roman Emperor Otto II married Theophano, the niece of the emperor John II Tzimiskes.
Ottomans: The second Ottoman emir called Orhan married the daughter of the emperor John VI Kantakouzenos, Theodora.
Russia: The Grand Prince Ivan III married Zoe/Sophia Palaiologina, the niece of the last Byzantine emperor Constantine XI.
So, the HRE had the earliest claim, but the Ottomans had the strongest one, after the fall of the Byzantine empire.
However, none of these states were true descendants of the Roman Empire. The Roman values and culture were long gone...
Invincibility lies in defence, while the possibility of victory in the attack -Sun Tzu
Akouson me, pataxon de (hit me, but first listen to me)-Themistocles to Euribiadis prior to the battle of Salamis.