Thanks, clears that up. :D
* Born March 1868 (per his headstone; the headstone inscription is more credible than the 1900 census)
* Turned 12 at Ellis Island (per family story)=
* He was at Ellis Island in March 1880.
* US Census conducted June 1880.
Also possible he immigrated 1881. I'm just saying don't count 1880 census out - as with the 1910 census, further investigation can turn up a different result than one's first thought. Keep an open mind.
Regardless of missing records, I think you're ready to go to the Sweden board asking for help to find his Swedish roots. You have much excellent information:
* a date of birth inscribed on his tombstone
* a family story re. his childhood that his parents died when he was young
* a family story re. his destination in USA
* his American name complete with middle initial
* his children's names (which are relevant)
* a close estimate of his immigration date (March 1880 or March 1881)
Only one person in Sweden would fit this constellation of characteristics.
May I offer another suggestion? As you present your research findings to the Sweden forum (if you do), I suggest you take care to avoid this type of statement: "My best estimate is 1881, so he wouldn't be on the 1880 Federal census." Really your best estimates (and mine) aren't genealogical fact. Note how previously I listed two immigration dates WITH THEIR SOURCES. Let the sources give THEIR best estimate.
It is a genealogical fact that a census record says a year of immigration. It is a genealogical fact that a family story says such and such. Let the sources do the talking.
The researchers who will help you are expert in assessing the credibility of various sources. They also are familiar with the variations and mistakes in records, which occur for many reasons.
The opening query says this: "My aunt believes that he had a birthday while on Ellis Island, which according to census records I've found would have been in 1877." To know the source in this sentence (the census source), we'd have to know which census records you've found, and which you haven't found. So even here you are the speaker, not the record itself. Clear yourself out of the way. Otherwise a "grapevine" effect happens - information gets a little distorted in every step away from the primary source.
Once your primary source gets to talk, you can then comment on the source, analyzing what it has to say.
Here's Princeton University definition of a primary (versus secondary) source.