Why did the pope name Charlemagne Emperor of the Roman people in 799?

We have no record of why Pope Leo III crowned Charlemagne as emperor. All we have is a record of the event and the fact that Charlemagne claimed he did not expect it to happen. The result is that the question is the subject of debate and differences of opinion.

I have read the view of some historians that the West was ready for the restoration of the Roman Empire, and had the belief that crowning Charlemagne emperor would facilitate this, to the benefit of all the people in Europe.

I have also read that Leo III may have believed the government of the Byzantine Empire, which never gave up calling itself the Roman Empire, was illegal because it was headed by a woman, Empress Irene. This leads to further speculation about what he intended to do with separate Roman Empires of the East and West, with the possibility that uniting them was envisioned.

And, of course, there are a number of other opinions, including the idea that Leo III merely wanted to express thanks that Charlemagne had saved him from a mob that had attacked him.
Nobody knows exactly why this is. Even so, Charlemagne was never really emperor. He was king of the Franks, a people who lived in modern day France and Germany. He lived 300 years after the true western half of Rome fell. The eastern half, Byzantium, already had a leader.