I think he’d still be eligible if his mother was a US citizen, even if he were born in the Phillipines. But maybe the birth certificate leaves off the name of the mother, or misspells it, or has some other ambiguity on that point.
Or maybe the birth certificate doesn’t exist.
Maybe the story that he was born in Hawaii was concocted because he discovered that birth certificates are not public documents in Hawaii.
As has been posted by others on this thread, when Obama was born (1961), a person born in a foreign country with one U.S. citizen parent and one non-citizen parent would be a U.S. citizen at birth only if the U.S. citizen parent had lived at least 10 years in the U.S. (which was clearly the case for Obama’s mother) and lived at least 5 years after the age of 16 in the U.S. (which was *not* the case for Obama’s mother, since he was born when his mother was only 20 years old). Thus, if Obama was not born in U.S. soil, he was not a U.S. citizen at birth, and thus would be ineligible to become president. (And if he didn’t subsequently become a naturalized citizen, he wouldn’t even be eligible to become a Senator or even to vote.)
That being said, I assume that Obama was indeed born in Hawaii and that he doesn’t want to release his birth certificate for another reason (such as because it says that his parents were not married to each other, or because it lists a second middle name).