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Ibrahim el-Houdaiby is exactly the sort of man the United States needs on its side these days. A 29-year-old graduate of the American University in Cairo, he is one of the firebrands who decamped to Tahrir Square, and he now opposes rule by both the Muslim Brotherhood and the Egyptian military. Houdaiby is an Islamist, not a Jeffersonian democrat. He believes passionately in sharia, or religious law, and the “totality of Islam” in governing the lives of believers like himself. In fact, he is Islamist royalty: His great-grandfather Hassan el-Houdaiby was once the brotherhood’s supreme leader, and his grandfather held the same post. But last year, young el-Houdaiby left the brotherhood in protest. Now he preaches a new, open kind of Islamism—one that doesn’t impose anything...