I'll lose the bonus points here, but if you have a very conservative court (say Reagan gets Bork and Souter turns out conservative) where O'Connor is far from being a swing vote and is essentially powerless, I could see O'Connor decide to try for the Presidency in 1992 or possibly be VP while being groomed to be President in 1996.
Hughes in 1916 is an obvious choice, but others I could see would be William O Douglas, Earl Warren, and Chief Justice Roberts (who would win the bonus points).
It would take an almost apocalyptic level of Watergate-level scandals in both parties executive and legislative branches for a "philosopher-king" type like sitting justice to run for and win the Presidency.