Vikings impact on Rome

[Nathan Ross]

What did the Frankish Kingdom of Italy call itself? 

I'm not sure! As far as I'm aware, the old Lombard kingdom was conquered by the Franks at the end of the 8th century and became a part of the Frankish Empire; apparently it was still referred to as a kingdom, although when or by whom, or in what language, I don't know.

As Charlemagne was apparently crowned Holy Roman Emperor in 800, I suppose we could refer to it as part of the Holy Roman Empire, although whether the inhabitants of that part of Italy would have used the term, or even have known of it, is another matter!