Despite mistaken impressions, Mr. Trump and his recent relatives had nothing to do with the surname change.

Mr. Oliver himself was careful to refer to a “prescient ancestor.” Indeed, several centuries have passed since Drumpf evolved into Trump, according to Gwenda Blair, who wrote a biography of Mr. Trump and his family. The Drumpf name appears in tax logs dating to about 1600, and the Trump name first appeared among his ancestors in Germany later that century, she said.

By the time Mr. Trump’s grandfather, Friedrich Trump, arrived in New York in 1885 at age 16 with a single suitcase and dreams of wealth, Trump was well-established as the family name, Ms. Blair said. In 1892, Friedrich Trump changed his name to Frederick Trump, a move toward Anglicization that was common among immigrants who hoped to accelerate their assimilation and fend off discrimination.

Immigrants in modern America, however, do not often change their names anymore. In June 2010, only about a half dozen of 500 applications for name changes in New York appeared to be Anglicizing their names, according to a New York Times analysis.

Name changes do remain popular among entertainers. For example, Jon Stewart, whose “Daily Show” is in some ways the parent of Mr. Oliver’s HBO program, was born Jonathan Stuart Leibowitz. As Mr. Oliver noted, Mr. Trump once criticized Mr. Stewart for changing his name, tweeting that “he should cherish his past — not run from it.”