Actually, under the law at that time, if Obama's father had been legally married to his mother, and O born outside the US, it would make no difference that his mother was a US citizen. At that time BOTH parents had to be US citizens if legally married. If the mother was unmarried, then her citizenship carried over to the child. But if the father was not married to the mother, and she was not a citizen, then neither was the child, even if the father was a citizen.
Here's the rule at that time for children born outside the US: married US citizen parents, child is a US citizen. Unmarried parents, child's citizenship follows the mother's citizenship. Obama is a US citizen because the marriage was void, since O Sr. was already married.