I know the Byzantine Empire was and would remain (historiographically speaking) the Eastern Roman Empire, and the Justinian dynasty was so close in time to the collapse of the Western Roman Empire. With the empire's early conquests, what specifically were Justinian I's plans for a restored Roman Empire? Was there a specific model he hoped to return it to--possibly something resembling Roman rule under Constantine?
This question seems very presentist. The idea of calling it the Byzantine empire attaches a modern distinction where there wasn't one in the ancient world.
The Byzantine Empire was not called as such in its time. It was called the Βασιλεία Ῥωμαίων or Imperium Romanum--the Roman Empire. To the Eastern Roman Empire under Justinian, there was no break from the previous empire, and they were merely reclaiming their lost territories.
Furthermore, the lines between East and West were not as clear as we tend to view them. The first division occurred under Diocletian at the turn of the 4th century. The Empire was reunified and divided several times before the traditional point at which we date the split of the Eastern and Western Empires--the split between the empires under Honorius and Arcadius in 395. While there was some sense that they were different entities for sure, especially as the East distanced itself from the West as the West fought Germanic invaders, the division was to an extent primarily administrative, and the Eastern and Western Empire were considered in their time to be the same political entity.