The main reason that isn't very helpful is that ignores the fact that the Eastern Roman Empire had parted with Latin altogether and had long since abandoned Italy by High Medieval times (usually said to have begun in 1066). It also ignores the fact that the Eastern Roman Empires needed to be called something during Medieval times and I don't see much of a historical prescedent of naming the political era you live in and call countries by that era. In Medieval times there were maps that called it the Roman Empire and maps that called it the Greek Empire. The term Byzantine being used for the entire empire did not come about until much later (referring to the original name of the area where Constantine's eastern capital was located, Byzantium) but it is still a fairly old term to refer to this empire so it's not like everybody would just switch over to something else overnight.

Besides, at this time there was still Rome itself, under the rule of the Pope, whose lands were known as the Papal States. Then there was the Holy Roman Emperor (originally just the Roman Emperor or Emperor of the Romans, the 'Holy' was never part of the actual title and onlt cane into use later on), which was originally an imperial title granted to Charlemagne by the Pope and vanished as a title when he died until his descendent Otto brought it back into use. After Otto's revival of the title, it became the name of the leader of what was basically a decentralized federation of German and Italian states loosley ruled by an emperor.

This is before even mentioning all of the other nations who claimed to be the continuation of the Roman Empire, including the Bulgarian, Serbian, Russian, and Turkish Empire. The Kingdom of Spain was declared in reference to Hispania, the historically important Roman province and the Kingdom of England and later Britain would do the same sort of thing by occasionally referring to itself as Britannia.

Any revision of the name Eastern Roman/Byzantine Empire is both unfeasible and really not very necessary. The people within this empire would seem just as alien to the subjects of Augustus as any other place that claimed to be the successor to the Roman Empire, or even the Roman Empire itself by the time of Constantine.

If you're interested in learning more about the Roman Empire itself, a podcast called 'The History of Rome' by Mike Duncan is a fantastic historical catalgoue.

Awesome question. Hopefully, I can give folks a background in the origin of these terms.

Interestingly, this has been a debate amongst Byzantine historians for a while now. The modern usage of the word as a term for the medieval, Greek-speaking Romans began in the Renaissance, when the humanist tradition took a renewed interest in the Roman and Greek traditions, in part because thousands of Byzantine citizens fled to Italy after the Ottoman capture of Constantinople in 1453 — bringing thousands of books that were rare or non-existent in the West.

In the 16th century, the scholar Hieronymus Wolf, in an attempt to differentiate between “Old Rome” in Italy and “New Rome” in Constantinople, began to call these Greek Romans “Byzantines,” as a reference to ancient town of Byzantion, which was absorbed in the 4th century into the city of Constantinople by the Emperor Constantine the Great. Louis XIV of 17th century France, an absolute monarch with a love for all things Byzantine, assembled dozens of scholars to build the Corpus Scriptorum Historiae Byzantinae, firmly cementing the term “Byzantine” into historiography. Edward Gibbon liked it too, albeit for a different reason, mostly involving Byzantine-bashing. Enlightenment-influenced Gibbon, in the The History of the Decline and Fall of the Roman Empire (volumes published 1776 to 1789), blamed Christianity in part for the “fall” of Rome. His book shook Byzantine historiography, and for much of the 19th century, historians shunned the Empire as “decadent” and “debauched” thanks to Gibbon. This is where the contemporary usage of Byzantine to describe complicated, overwrought bureaucracy and intrigue originated.

Back to the actual time of the Empire. For most of the early/middle timeline, the Emperor in Constantinople was the Emperor of the Romans. Up to the year 800, Christendom had in theory been under the symbolic leadership of the Emperor in Constantinople. Even the off and on Western enemies of the Empire, such as the Ostrogoths, Franks, and Vandals, were in theory “patricians” of the Roman Empire, who ruled in the West by imperial sanction. At this time of uncertainty, or as some historians call it, the Age of Anxiety, the symbols of Church and Emperor were enduring and in many ways integrated with one another. Since the time of Constantine, the Emperor in the East had played a key role in shaping the Church. However, that all changed with the first and most important schism between the Western and Eastern churches: Iconoclasm. Suffice it to say, Iconoclasts didn’t like religious images (idolatry) and Iconodules venerated images as conduits to the divine (Ikons).

With the combination of iconoclasm, which did not endear the Bishop of Rome, Stephen, to the Emperor, and an imminent Lombard invasion, Stephen appealed to the powerful Frankish kingdom for aid. Pepin the Short answered. Afterwards, the Emperor granted him the title Patrician — just like the earlier Ostrogoths and Vandals.

Then Christmas 800 A.D. happened. Pepin’s successor, Charlemagne, was praying quietly in a chapel, and along came the Bishop of Rome, who crowned him Emperor of the Romans. This was a huge deal. In an age of symbolism, titles meant a lot. To Europeans at this point, there could only be one Christian Roman emperor… who was the true emperor? This divide caused massive damage to East-West relations. The Emperor in Constantinople repudiated the Bishop of Rome. It remained a point of contention for the rest of history (look at every single entity claiming to be the heirs of Rome, from the German Holy Roman Empire to Tsarist Russia to Nazi Germany). People have always been obsessed with Rome and the supposed legitimacy it confers.

From this time forward, some in the West began to refer to the East as the Empire of the Greeks, and its citizens as the Greek Romans. Meanwhile, the Byzantines never changed their attitude: they were Romaioi, a Greek translation of Romans. There is some evidence to suggest that residents of the capitol might call themselves “Byzantines,” but the Empire, even to its last days, was considered Roman and Christian. Rulers in the West, as late as the final century of the Empire, would still refer to the ruler in Constantinople as a (and sometimes, the) Roman Emperor.

Hope that’s informative.

As for the benefit of terminology: I believe that the term Byzantine is good for distinction's sake. The Byzantine Empire was truly a unique one, with a Roman heritage, Greek language, Medieval context, and heavily Christian outlook. Even if the historical citizens saw themselves as the heirs of the Romans, they would identify as Christian first, which distinguishes them from the old pagan Rome.

Sources: Edward Gibbon (The Decline and Fall of the Roman Empire) David Proctor, Tufts University (Professor) Steven Marrone, Tufts University (Professor) John Norwich (A Short History of Byzantium) Various French Scholars (Corpus Scriptorum Historiae Byzantinae).

Props to /u/B-Hosk for an excellent answer. I would like to develop this a little more though. Thankfully, B-Hosk covered a lot of ground and that I don't need to cover.

There is a lot more at stake historiographically than some other posters have discussed. While the term Byzantine Empire is indeed from the 16th century as /u/B-Hosk mentions, it's power, I think, emerged later. In order to understand the gravity of this question, we need to look back to the origins of medieval studies in the 19th century or so. Medieval history emerged as a separate subfield as a thoroughly nationalist exercise. The goal of medieval history at that time was to find the origins of modern European nations. Just look up ethnogenesis and get blown away. There's a nice article by Walter Pole in Debating the Middle Ages edited by Lester Little and Barbara Rosenwein (both stellar scholars) about conceptions of ethnicity in the Early Middle Ages. Anyone interested in medieval historiography should really pick this book up.

The other thing to recognize about this question is that since the Middle Ages, Rome has been considered the apogee of Western culture. With this in mind, if one locates the origins of modern European nations in the Germanic tribes that invaded the empire, that would suggest that these nations destroyed Rome. The solution to this issue was to position the Germanic successor states as exactly that, the heirs to the Roman legacy. This attitude is at the heart of Pirenne's Mahomet et Charlemagne. The unity of the Mediterranean wasn't destroyed by the germanic tribes, it was the forces of Islam. The Germans preserved the Roman Empire.

To call the Byzantine Empire, Roman is, in a sense, a denial that medieval Latin Europe was not the true heir to the Empire.

With this in mind, in graduate school work, I refer to them as the Roman Empire. My (elderly old school adviser) hasn't raised an eyebrow over the issue. It's what the they referred to themselves as, and they were described as such by contemporary sources. They weren't the Empire of the Greeks in the records I've seen, they're the Romans.

One last point I'd like to make is that Michel Balard, one of the finest scholars of the medieval Mediterranean referred to the Byzantine Empire as the Roman Empire. Not something to be taken likely.