https://bit.ly/2s1uWV0

If you have a date of entry, you can look through the people who arrived that day.

Also, if you have any old documents or letters (like from relatives back in the Old Country), you can maybe find a town or village name.

TransAtlantic cable was in operation by 1872. Ship's manifest were cabled to ports of entry so as to compare for known anarchists. It was also known within hours, when the ship was to reach port. At that time, the immigration authorities would have people who knew the languages and alpabetic characters, on standby.

As the years pass, it's easy to get into '72, '74 whatever it takes. If you haven't found census records close to when they immigrated, broaden your search on their names, but in the same general area in older censuses. Remember that the answers are relevant to the day of the census, i.e. "now", the census is not always on the same day of the year. People sometime shave a few years off as they get older, or claim they have been here longer.

The "clerk at the dock" rarely, if ever happened. When people got on the ships the manifest was filled out by people that spoke the language. When they got here, the manifests were used - no "clerk at the dock who didn't speak the language" was likely involved. People generally didn't change their name until they were in their new country for a bit. After they had been there awhile, they changed their names to fit in better.

I have ancestors that decided Olsson was too common and used their grandfather's name as the basis of what became my last name rather than their father's name. Others changed the spelling. Norwegian has vowels that don't exist in English and some consonants that are pronounced differently. Hvaal becomes Wall, Wahl or Hall. Qvam naturally becomes Quam. Skjerven becomes Sherven.

I have a World Explorer subscription to Ancestry.com. If you want, you can PM the details you have and I can take a look. Minnesota and Scandinavia, particularly Norway is my usual turf, but I can take a look.