What is amazing is that there seems to be no real mechanism to determine eligibility. Theoretically, the Democratic party could run someone who moved to the USA at 6 months old, but has claimed to be born in the USA, jiggle the paperwork and get all the way to the electors. As long as he wasn’t a known person (like Arnold S.) you obviously could take it all the way to at least this point. If he was popular enough it is doubtful anything would come of it.
BESIDES the point of Obama, this is a considerable problem area in the future.