We have all heard someone say that their family name was "changed by the inspectors at Ellis Island." Nowadays our names are recorded when we are born and are virtually never changed. You can still use any name you want as long as you do not intend to defraud but, in fact, with drivers' licenses, social security numbers, credit cards, etc., it is just too complicated to try to alter your name except through a court proceeding.
People seem to feel that it was the same way at the turn of the century. They think that immigrants had one correct way to spell their name in the old country, when they encountered the clerk at Ellis Island it was changed to something else and then it was spelled that way ever after in America. The explanation usually is that the immigrant spoke little or no English, so either the immigrant inadvertently gave an incorrect reply to the question of "What is your name?" or the clerk misunderstood the name or decided it was too complicated.
In reality, it is highly unlikely that this happened. The Immigration and Naturalization Service has a good article on immigrant name changes that explains why this wonderful story is a myth: the clerks at Ellis Island didn't write down names. They worked from lists that were created by the shipping companies. What usually happened was the emigrant bought a ticket from an office near his home. So, the seller probably spoke the same language and transcribed the name correctly. In cases where the name was recorded incorrectly, it likely occurred in the old country, not at Ellis Island...