Yes, Ted Cruz Can Legally Become President
I get the fun irony of tossing eligibility questions at the Senator from Texas, but let's settle that now: As David Graham writes at the Atlantic:
[T]he Congressional Research Service gathered all of the information relevant to Cruz's case a few years ago, at the height (nadir?) of Obama birtherism.
In short, the Constitution says that the president must be a natural-born citizen. "The weight of scholarly legal and historical opinion appears to support the notion that 'natural born Citizen' means one who is entitled under the Constitution or laws of the United States to U.S. citizenship 'at birth' or 'by birth,' including any child born 'in' the United States, the children of United States citizens born abroad, and those born abroad of one citizen parents who has met U.S. residency requirements," the CRS's Jack Maskell wrote.
So in short: Cruz is a citizen; Cruz is not naturalized; therefore Cruz is a natural-born citizen, and in any case his mother is a citizen.