My grandfather came to NY around 1900 (haven't found his arrival) and sent for my grandmother and four children in about 1901 (haven't found theirs either). My mother told me that he "told [her mother] what name to use." The name was FEINBERG and that is the name that appears on his death certificate where it asks for his father's name.
I asked my mother what the original name was and she said something that sounded like "DYEBUCK". I have found evidence that the family came >from the area of Brest-Litovsk and I have seen the name DAJBOG on JewishGen lists for that area, so I think this may be the name. But so far it's a dead end in terms of matching anything up.
I have wondered how people managed to obtain passports under these circumstances. Some of the stories on this list have been very informative.
