The Hammerstein Ballroom is actually named for Oscar Hammerstein I—who constructed it as the Manhattan Opera House in 1906—not his far more famous grandson, Oscar Hammerstein II, of the duo Rodgers and Hammerstein, who wrote Carousel, South Pacific, The King and I, Oklahoma! and The Sound of Music, and who died in 1960.

Simon Hammerstein is that Oscar Hammerstein’s grandson, but says it’s the first Oscar Hammerstein’s vision he looks to for inspiration.

“I’d like to think I’m following in his footsteps, actually. He was the man,” Mr. Hammerstein said. “He was 11 years old and in Germany when his mother died—he inherited his love of opera from her—and his father was abusive. So he took his violin and sold it for a ticket to America. Isn’t that amazing?”

As Simon Hammerstein turns 29 next week, he’s preparing to open The Box—a modern take on the old-fashioned supper club—in February.

When Oscar Hammerstein (the first) arrived in America in 1863, he got a job sweeping floors at a cigar factory before eventually taking out a patent on a machine that could roll cigars. He built the Harlem Opera House in 1889 and, to lure the crowds away from the Metropolitan Opera House, built an opera house in Manhattan. “They eventually bribed him to stop competing,” said Mr. Hammerstein. “They gave him a million dollars—which was a fortune at the turn of the century. He went to London and built the London Opera House.”

Mr. Hammerstein, who is single and lives in a loft on the Lower East Side, was giving a reporter a tour of The Box, a 160-seat venue in a 5,000-square-foot former sign factory on Chrystie Street. He was wearing a cream sweater under a chocolate-colored blazer with tan trousers and sneakers, a cigarette dangling from his lips. He sported a full mop of dark, unruly hair and a youthful beard partly concealing a sheepish, yet also mischievous, smile. A budding theater director, Mr. Hammerstein is also a nightlife impresario in the making.

“One day it dawned on me that being a director is having your whole life depend on what The New York Times says about you,” he said. “It’s not a life.”

His first venture is thick with mood: The Box contains nooks and crannies filled with 1920’s artifacts; the walls are covered in mismatched vintage wallpaper; the small stage is draped with a heavy mustard-colored velvet curtain. “I want it to look as if you’re in the home of someone a bit eccentric,” he said.

Born in England, Mr. Hammerstein has an accent that dips between posh Brit and American. (“I’d never be an actor—I mumble too much.”) Growing up with the Hammerstein surname, he said, was less pressure for him than it was for his father, James.

“My father had those big shoes to fill,” he said. “That was sort of overwhelming for his generation.” James Hammerstein had followed his father into the theater, acting as the stage manager on the first-ever production of Damn Yankees and then directing much of his father’s work. He died in 1999. “My father was very thoughtful and smart and took theater very seriously,” said Mr. Hammerstein. “He loved seeing shows and then talking about them afterwards.”

James Hammerstein, along with his wife, the British playwright and actor Geraldine Sherman, would take young Simon to Broadway. “I was jealous of my parents,” he said. “They were much cooler than me. Normally, one is embarrassed by your parents. Not me. Any girl I was interested in would end up chatting with my mom.”

Mr. Hammerstein left school at 16 and found work as a stagehand, working his way into directing. “My father was very classy; I had more of a punk attitude,” he said. As for his famous grandfather, Mr. Hammerstein didn’t get too much inside scoop.

“I guess he wrote all the time, so he was never really home,” he said. “I don’t think he got the hang of children. My father didn’t regale me with stories of his childhood too much. It was always a bit of a mystery.”

One story that he does know is that James Hammerstein’s third birthday party was covered by Time magazine, and he was chauffeured to kindergarten. But the glamour didn’t stick.

“My father hated Hollywood,” said Simon. “He was so not bling like that—he drove a crappy Mazda!”

When it came to his own leanings toward theater, Mr. Hammerstein tried not to think too much about his bloodline: “I felt like directing was something I was talented at, and it pleased me that I might have some sort of natural connection to it.”

As for carrying on the family name, Mr. Hammerstein said he was conscious of the conclusions that others might make. “I do get paranoid that people are leering at me, like, ‘Oh, Daddy got you a job.’ Though I suppose they can’t say that anymore. I try not to think about it; I don’t want to look back at my life and think I exploited anything.”

But he couldn’t help lighting up when talking about the “dinner circus” that would take place on his stage.

“I want it unpretentious,” he said. “I want it to be a place where people can eat, drink, talk, text, kiss—whatever—as well as be entertained. I do get a sense of joy when the audience is breathing in the same rhythm.”