There's no way for Greece to have any long term control over even Izmir in Anatolia, short of some nightmare Nazi victory sort of scenario where a Nazi aligned Greece is given the resources and permission to conduct ethnic cleansing/genocide on the native Turks, but Thrace is possible in a less terrible world. Perhaps Russia gets what it wants in the Balkans but instead of taking Constantinople themselves to give it to an allied/puppet Greece to mitigate British displeasure, or something along those lines. Greece taking the territory under its own power is less likely, almost impossible with a stable Anatolian based state, but given the zaniness of the Near East a century ago it's probably doable under the right circumstances.
The more difficult question is having the Greeks rename their country by a name that they've never used themselves. The Greeks today refer to themselves as the "Hellenic Republic", and appear to identify more with the ancient Hellenes as opposed to the medieval Rhomaoi. It's difficult to see how the Greeks would develop an identity with discredited Rhomania (which given the time frame we're talking about was perceived by most western scholars as a decadent, even degenerate perversion of Ancient Rome), and even more so for them to label themselves with the very label which carries with it such connotations. A different Greek war of independence could possibly result in a Greece which thought of itself as the current iteration of Roman civilization, but this won't occur from any OTL foreign aide, as unless I'm mistaken there're no states in Europe which want a small impoverished peninsula to take on the prestige of Ancient Rome; they wanted it for themselves. At the same time, while a war of independence without any foreign aide is surely possible with the right circumstances, for the victorious Greeks to then use the frankly racist term "Byzantine" as opposed to "Rhomanian" to name their state and nation seems difficult to imagine. Ultimately the term "Byzantine" needs to have markedly different meanings to OTL for such a scenario to eventuate.