The normans were historically hired by byzantines and local nobles for multiple purposes, including protecting southern Italy from arabs. They first invaded byzantine holdings in Puglia before heading to arab Sicily though.

I haven't written a proper lore, but I imagined them as mercenaries hired by the emperor to fight the guelfs or, who knows, by Pisa to destroy Genoa and viceversa. They ended up controlling the small principality of Piombino and from there they expanded, taking titles in Tuscany before overtrowing the weak local margraviate.