Li Rosi is decidedly Sicilian, of the Catanese in particular of Vizzini, Catania, Grammichele and Licodia Eubea, it should derive from the word rosa (flower) to indicate the presence of perhaps a large rose garden near the family home.
Rosa is absolutely pan-Italian, Rosi is more concentrated in the North center, should derive from the name Rosa.
Diffused in central Italy where it could descend from the toponym Rosia (SI), in Emilia and in Milan.
You may be interested
Verba Volant, Scripta Manent