I had a similar experience trying to find a relative. I was fortunate enough, after many, many searches, to find his name in the Naturalization database on Ancestry.com. Of course his name was misspelled beyond belief. I discovered he arrived in the US under an alias, that's why I couldn't find him.
Normally one would find the ancestor on the ship's manifest, then the census records to see if he was naturalized, and then order the naturalization papers. I did the opposite. I found him first, by luck, in the naturalization database so when I ordered his papers I was able to find his alias, then I found him in the Ellis Island database.
I would ask your relatives if anyone might have your great grandfather's naturalization papers or other documents. Over the years, I have found that relatives have boxes of photographs and documents in the attic that they forgot they had, etc. So ask your relatives, you might be pleasantly surprised. Chances are your relatives took the misspelled last name off their immigration papers.
This website might be of help. You can search by first name only, place of birth, date of birth, etc., or any combination. It's for
ellis island only. Good luck.
http://stevemorse.org/