One place to start collecting immigration songs would be Jerry Silverman's collection The Yiddish Song Book. Here are some songs in the section entitled "Amerike," including quite a few about the immigrant experience after going through Ellis Island. These are Jewish songs, but no doubt the experience was similar for others.

Lidl fun Goldenem Land (Song of the Golden Land) by Mordechai Gebirtig
Shikt a Tiket (Send a Ticket)
Kesl Gardn (Castle Garden) by Morris Rosenfeld and Mark Warshawsky (Castle Garden was the processing center for immigrants before Ellis Island opened.)
Ellis Ayland (Ellis Island)
New Yorker Trern (The Tears of New York)
Mayn Rue Plats (My Resting Place) by Morris Rosenfeld -- a hauntingly beautiful song, one of my favorites
Mayn Yingele (My Little Son) by Morris Rosenfeld
Svetshop (Sweatshop) by David Edelstadt
Arbeter Froyen (You Working Women) by David Edelstadt
A Briv Fun Amerike (A Letter from America) by Mark Warshawsky

Aloha,
Mark