Theodore B. Olson’s office is a testament to his iconic status in the conservative legal movement. A framed photograph of Ronald Reagan, the first of two Republican presidents Mr. Olson served, is warmly inscribed with “heartfelt thanks.” Fifty-five white quills commemorate each of his appearances before the Supreme Court, where he most famously argued the 2000 election case that put George W. Bush in the White House. On the bookshelf sits a Defense Department medal honoring his legal defense of Mr. Bush’s counterterrorism policies after Sept. 11.

But in a war room down the hall, where Mr. Olson is preparing for what he believes could be the most important case of his career, the binders stuffed with briefs, case law and notes offer a different take on a man many liberals love to hate. They are filled with arguments Mr. Olson hopes will lead to a Supreme Court decision with the potential to reshape the legal and social landscape along the lines of cases like Brown v. Board of Education and Roe v. Wade: the legalization of same-sex marriage nationwide.