You won't. Because it NEVER HAPPENED.
Certainly, lots and lots of immigrants changed their names after living in the US for a while. The most common time is when they naturalized, and this is usually noted on the naturalization papers. They can do it at other times as well. My Polish grandfather changed his surname as an adult (he naturalized on his mother's papers when he was younger).
What never happened is changes when someone arrived. Ellis Island, Castle Garden, etc. Never ever. No matter how many family stories you have. Now, you didn't say your family story was at the point of immigration, maybe it's later. Just that it implied it was at immigration ("on arrival").
Can you tell this is a pet peeve of mine? :-)
My husband's family had a story that his paternal grandfather's family had their name changed at Ellis Island. From Nitowitz to Norwitz. I have no idea why they got that idea or how the myth persisted because it was amazingly easy for me to debunk it. But then the records that are so easy for me to access weren't online 20 years ago and required microfilm searches in the area where they were generated.
My grandfather-in-law was born on the boat in 1899 (that family story is true! I have the proof) and seemed to be considered an American citizen at birth (I can't confirm this because I don't have a birth certificate but everything points to it). He and his family were Nitowitz (with a spelling variant) on the boat, through Ellis Island, and in the census and other records for decades. At some point after he became an adult, my grandfather-in-law became Norwitz. But his parents and siblings all stuck with Nitowitz. Myth busted!
For your relative, I suggest finding as many records as you can for him. Include things that might look like they're not that important, like city directory (and phone book) records. When you come across something, file it under the name it gives (full name...middle initials and etc can be important) and note the source. I can do this with ease in Family Tree Maker but it's a little harder on Ancestry because it doesn't let you specify sources for specific facts. Eventually, you will find the place where his name changed and you can narrow down the timeframe. Note that it's likely he used his new name in a few places before officially changing it, so don't assume the official change is before the first use. Also, he may never have changed it officially. That happens sometimes, especially in the days before computer records.
I also wonder what the story is behind your relative's name change. Frans to Frank follows the pattern of keeping the name but Americanizing or simplifying it. But Fredlund is nothing like Anderson and Anderson is a common American name whereas Fredlund is more unusual. So there may be something else going on. Like a stepfather in the picture, or him living with another family, or him trying to hide his identity (this last one is the least likely but the most interesting).
I'd start with the 1900 Census and work your way up and down from there.
Good luck with it,
Cyndi