In many ways, what I am proposing about Joseph Smith having Amerindian ancestry is likely to ruffle some feathers, because so much is already known about his ancestry. Mormons are famous for doing genealogy; it would be somewhat astonishing to not know such an essential fact about its leader's history. Or more interestingly, to know and keep it secret....
Anyway, to kind of keep myself in check, and to find out more about what is known, I went to the library and found a book called "Joseph Smith's New England Heritage" by Richard Lloyd Anderson. I'm mostly skimming the book to find more strictly biographical details.
Here is a pedigree chart from the book, detailing Joseph Smith's maternal line:
Nothing particularly native sounding. "Mack" is said to trace to Scotland on his paternal side. Some of the information here is a little on the sketchy side, as in the trail ends sooner than on other lines. I have seen one forum commenter on a genealogy board stating something about Solomon Mack being secretive about his genealogy, but she did not elaborate. I have yet to figure out what she was referring to.
Ebenezer Mack's mother, Sarah Bagley (1663-1720), has very early Massachusetts ancestry, but it all appears to originate in England. My records (pilfered as they are) suggest some members of this line in Massachussets in the early 1600's -- well before America was an independent country.
The Hannah Huntley line appears to be mostly English in origin as well, although she does have a Balthasar DeWolf in her line who came from either Poland or Germany, and he at least may have been Jewish. More on that later (a lot more!). Lots of Connecticut and Massachusetts.
I have less information on the Lydia Gates line, but much of what I have states it again traces to England. A few lines dead end for me in Massachussetts and Connecticut, but names seem to suggest more of the same -- England.
So, if we are to suppose that it is possible that Joseph Smith had some kind of Amerindian ancestry through his mother, the most likely candidate for that position would be Ebenezer Mack. None of the other lines appear likely to have any Native ancestry for any reason, if the records are accurate. Even if they did, it would be very difficult to measure with any accuracy through DNA testing where it is now, since autosomal DNA signals are not as strong after about five generations. Plus DNA testing has no easy way now of separating your ancestors to find out who contributed what.
It is more like me without my glasses trying to guess who a person approaching me is and making multiple simplified guesses. It gives an interesting idea of who I am and who I am like, but really isn't that great for teasing out single ancestors from multiple generations back. Although you can find people who match you on certain segments of DNA, and if those people are willing to respond to you AND they really know their family history well, you may be able to make some stronger conclusions.