To teach the potential marketing benefits to rebranding, business schools often turn to the fascinating tale of the Patagonian toothfish.

In the late 1970s, a Los Angeles seafood merchant named Lee Lantz happened upon the Patagonian toothfish while shopping at a coastal fish market in southern Chile. Known by locals as “bacalao de profundidad” (“cod of the deep” in English), the ugly fish with the ugly name was relatively obscure at the time.

In fact, many local fishermen thought the cold-water ...