Was it really very common for immigration authorities at Ellis Island to change or mix up the names of immigrants in the past?
I've heard several stories from friends about their last name resulting from mixups at Ellis Island at the time of their immigration to the United States. My own last name is the name of a town in another country, for instance, but is not a common last name in that country. It seems likely it was inadvertently or deliberately changed. Did this happen a lot, and if so, why? When was it fixed?
tl;dr: Names weren't mixed up at Ellis Island. It's a pervasive myth, and a bit part of family oral histories, but not something that was actually happening in any significant degree (if it was happening even at all, which there's not evidence to suggest).
The process was this:
You get your tickets / arrange passage in your country of origin. At that time, the ship's manifest (list of all passengers) is compiled. In most cases the person composing the manifest will be a speaker of your language, unless you've travelled far to get to that point (totally possible). Regardless, it was your name as recorded on the manifest that the immigration agents at Ellis Island then received. In certain cases names were misspelled on the manifest (i.e. not Ellis Island's fault) and in certain cases mistakes were corrected by agents at Ellis Island. In rare cases where your name was wrong on the manifest and not corrected, this still wouldn't have been a big deal. The only person it would matter to would be your descendants who might have trouble locating records, because once you started your life in America, the way your name was spelt was in your own hands.
Did people change their names? Absolutely. Did those names get changed because some clerk at Immigration was a buffoon? Nope, and even if they were a buffoon and even if that were cause to have your name messed up (it wasn't), it was completely non-binding.
In other words, there simply was no room for the mixup. If your name ended up being different it was because you yourself made that happen, which many people chose to do.
This also applies for other immigration operations such as Angel Island, about which I've written quite a bit here in the past as it relates to the Chinese Exclusion period. If you want names that an Anglo-American would have trouble with but still didn't mess up (because manifests), that's a good place to look.
If your grandma is telling you your name is the way it is because Ellis Island messed up, she is almost certainly mistaken. But that's okay; family mythologies are still important.
For easy reads on the topic, the Smithsonian and NYPL both had well-done articles on this last year, available here:
Ellis Island Isn’t to Blame for Your Family’s Name Change (Smithsonian)
Why Your Family Name Was Not Changed at Ellis Island (and One That Was) (New York Public Library)
OP: I've already commented here on this, but not as a top level comment, so I thought I'd consolidate what I've already written into this comment in case people stumble across this later, (and also because people get cranky when posts don't have a top level answer)