Laura I was able to find my maternal grandparents who came here between 1909 and 1913 from Aleppo Syria (Haleb), Syrian Catholics.I did so via Ellis Island Web site, but it is difficult. You must use the fuzzy search to find sound alike names, and then look at the ship manifests themselves on line, as the Mormon volunteers who keyed these in compounded the errors from the original immigration officers.
As stated by others, many times the immigrant's last point of departure such as Alexandria or Naples is listed instead of their real country of origin, which often does show correctly in the digitized images of the handwritten manifest records on file at this useful Web site. If you know the dates they came in on or ship name, and just search the manifests, this can help too if you cannot find a name match.
By the way, my paternal grandfather (who I cannot yet find) was named Rizullah Shagawat, who came to US during this same period. Rizullah is believed to mean "Prince of God" in Arabic.