SEOUL, South Korea — Plenty is off limits to North Koreans. Televisions that receive anything other than government broadcasts. Traveling outside their hometowns without official permission. And Bibles.

Add the name Jong-un to the list.

The given name of North Korea’s young and capricious leader, Kim Jong-un, now appears to be reserved only for him, carrying on a tradition started by his grandfather, the founder of one of the world’s most brutal police states. And those North Koreans who had the name when the young Mr. Kim came to power in 2011 have had to give it up.

Such is the well-honed cult of personality in North Korea, where the leader is something of a godlike figure and where critics of Mr. Kim can find themselves in the nation’s notorious gulags. For a time, early in Mr. Kim’s rule, outside analysts and foreign diplomats held out hope that he might be more modern and open than his grandfather and father, the country’s first two leaders.

But Mr. Kim has proved to be no soft touch: He had his uncle and longtime mentor executed by a firing squad; he was accused of plotting a coup. Mr. Kim has also thumbed his nose at not only the United States and South Korea, his country’s longtime nemeses, but also his Chinese benefactors, who have pleaded for less provocative behavior following nuclear weapon and missile tests.