Donald Trump is set to deliver his acceptance speech for the Republican nomination Thursday night in Cleveland. Behind his words will be 30-year-old policy adviser Stephen Miller, who is tasked with drafting the speech of his life.

If it weren’t difficult enough to produce a blockbuster address for the most unpredictable Republican nominee in modern memory, Mr. Miller will need to shoulder the additional scrutiny foisted on the candidate after his campaign admitted on Wednesday that parts of the convention speech his wife...