Byzantine, Islamic, and Frankish worlds had replaced what was the Roman Empire. Which state most deserves the title of Rome’s successor and why?

Intriguing question. Each of these three "worlds" can be argued to have replaced a part of the Roman Empire. In the end none of them replaced it completely.
The Byzantine or Eastern Roman Empire never "replaced" the greater Roman Empire, but it continued aspects of its heritage for some time to come. Its height was during the reign of Justinian I, not long after the fall of Rome (ca. mid-6th c. BCE). Thereafter the Byzantine Empire remained solvent but was constantly defending itself from Islamic incursions from the later middle ages on, and it finally fell to the Ottomans in the mid-15th c. While one could say it was the best candidate to be a Roman successor, this is complicated by some things. For one, Orthodox Christianity was quite different from Catholicism.
The Islamic Empire expanded rapidly after the death of Muhammad. The initial Caliphates took control of the Arabian Peninsula, Asia Minor, and North Africa in rapid succession. The Umayyad Caliphate was the largest (this was the period when Muslims captured Spain from the Visigoths), but the Abbasid Caliphate was the most stable Islamic Empire, existing from the mid-8th to the mid-13th century. Again, while these Islamic empires were impressive in their extent, none of them captured Europe. And none of them, obviously, were Christian.
The Franks, while they ruled the smallest of the three Empires mentioned here, have perhaps the strongest claim to replacing the Roman legacy. The most elaborate expression of Frankish rule was the Carolingian Empire, first established by Charlemagne, who was actually crowned the "Holy Roman Emperor" by the Pope in 800 CE. While the Carolingian Empire didn't exist on the scale of the Byzantine or Islamic Empires, it continued Roman traditions most extensively. Charlemagne built churches in the Romanesque style, converted pagans to the Christian faith at the point of a sword, and emphasized the value of education. If, however, one considers which culture best preserved the legacy of Greek and Roman thought and amended that thought as well, then title goes to the Islamic Empire. Before it fell in the mid-13th c., Baghdad was a major cultural center—not rivaling Rome, but nonetheless significant.