The only information I was able to find for your great grandmother was her passenger log from when she arrived in Manhattan with her children.http://search.ancestry.com/iexec/?htx=View&r=an&dbid...
My family also came from Castellammare to Manhattan around that time and later moved to Brooklyn. There are only seven Camardas in Castellammare now, so it seems as though they are a small family and it is very likely that we are related. However, I wasn't able to find any proof.
A decade later, they would ask for information on who you were going to live with, as well as who your next of kin was back in your old country. However, when Dora came to NY, they only asked who was sponsoring her, and she answered her husband and his address in Manhattan. Not very helpful, I'm afraid.