These are names I have heard associated with my last name. I didn't find any of them listed, so I'm wondering... how can I determine not only origin, but if certain family stories are true? I've always been told that Stasio was a shortened version of one of the other two... done so by my great grandfather because he did not want to be associated with other relatives who had come to the US.

In order to do anything you need to trace backwards starting with yourself, then your parents and then you grandfather. At some point you need to use the info you find to locate his ship manifest. To help narrow it down do you know where in Italy he came from? Does your family speak any Italian? A dialect perhaps to help narrow things down...