Left out the last line- The Smithsonian article seems to be close to the truth.

As near as I can find out, Santa Anna's antecedents used that spelling for some time. The saint's name Santa Ana, does not seem to be a factor, except possibly back in time.
There are a number of living Americans, and 19th c. records of Americans, with the last name Santa Anna. I am sure that Spanish records would turn up many with two n's.'