It has not been an easy few months for Judith Giuliani. Her rollout to the public received rocky reviews from the political class, Republicans included. A series of negative articles about her shopping habits, marital past and supposedly testy relations with campaign staff followed. Her appearances alongside her husband, Rudolph W. Giuliani, grew suddenly scarce — and some analysts suggested that she keep it that way. So it was perhaps no surprise that at a recent lunch in downtown Manhattan, Mrs. Giuliani offered this self-assessment: “When it comes to politics, I’m new to this.” Over the course of a two-hour...