Nate Thoma stood up in a Delaware bankruptcy court last December in a sharkskin suit and delivered a 24-minute argument that changed the course of one of the largest bankruptcies in U.S. history.

The 33-year-old Washington Mutual investor, with no legal experience, delivered what people in the courtroom called an unusually eloquent speech, helping persuade the judge to investigate trading by some of the nation's biggest hedge funds and to reject a plan for the bank's exit from bankruptcy.

The...