Also what would surrounding countries have referred to them as?

The Arabs called them "the Romans". The Qur'an, for example, has a verse that says "the Romans have been defeated" (ghulibati ar-rum), by which the text means the Byzantine Empire. That's also why the Sultanate of Rum (i.e. Rome) that existed from the 11th to the 14th centuries was entirely contained within Anatolia. To this day, the Greek Orthodox of Lebanon have a (half) joking expression "Allah Rumi" "God is Roman", by which they mean Greek Orthodox, not Roman Catholic.

This is a complicated question so there's always more than can be said, but I've answered a couple of similar questions in the past, which might help:

What did the other people's of Europe call the Byzantine Empire? Did it differ depending on who you asked and were there any changes as the empire dwindled?

Why is it called the Byzantine Empire when the city was known as Constantinople for centuries longer than it was called Byzantium?