Law profs Lawrence Tribe and Mary Brigid McManamon have convinced me.
By the intended meaning of the U.S. Constitution's requirement for a "natural born Citizen", Ted Cruz is not legally eligible to be President.
McManamon explains the distinction. A natural-born citizen is one born in U.S. territory. (Frustratingly, McManamon doesn't clarify whether this is restricted to the metropolitan U.S. or includes other territory under U.S. control, which would cover Barry Goldwater and John McCain.) Cruz wasn't. He was born in a foreign country. End of story.
But his mother was a U.S. citizen, you object. Here's where McManamon gets ingenious. His mother's status made Cruz a naturalized citizen: naturalized automatically, at birth, instead of having to go through a naturalization procedure later on. But he's not a natural-born citizen.
That's what the Constitution was written to intend. Tribe puts the argument in context. If put to the Court today, the decision would probably go for the more inclusive meaning. But Cruz wants justices who stick to the original intent. Well, then, he can't have them, because they'd have to rule he was ineligible to appoint them.
I have no doubt, however, that if Cruz is nominated and then elected, the question will be just brushed aside.