Cherso is the Italian name for the town and island of Cres (pronounced "Tsres" in Slavic), which is now a part of Croatia. Cherso was ruled by the Roman Empire and the succeeding Byzantine Empire until the Croats took over in the 7th century. The island eventually came under rule of the Venetian Republic in the 10th century, to which it belong (for the most part) until 1798. Austria controlled the island until the end of the first World War, when Cherso became an Italian island again until 1947, at which point the island (and the rest of Istria) was assigned to Croatia.
Cherso has always been a key port for trade, the reason its inhabitants' ethnicities are split between Italian and Slavic descents. Many of the residents speak both Italian and Croatian.
Some more information on the "Chersini" immigrants in the US at http://www.societychersina.com/index.html .
And, thanks to a Rodinis cousin Emil Udovich, I can show you where the Palazzo Rodinis was located (indicated by the green marking on the inset):
