The folks at Ellis Island were responsible for checking ship manifests. They didn’t have any paperwork to change people’s names, nor did were there any laws requiring anybody to do so. At the time, people in New York could legally change the spelling of their name just by using a new spelling though, which probably helps account for all the name changes that happened.
The only time names ever really changed at Ellis Island is if the immigrant decided to change their name to fit in with American society or the local community to which they were moving. Sometimes, they changed their name before even getting on the ship to come to America. There was no requirement to immigrants to use their real last name when coming to America. They could give themselves any name they liked. As long as the name they gave matched up with the answers they gave according to their profile on the ship’s manifest, they were good to enter the nation and use the name they chose.