Joan Bingham’s motherly instincts kicked in when I joked that her son, Robert Bingham, had attended a less selective college than his wife, Vanessa Chase Lilly. Ms. Lilly attended Harvard. Mr. Bingham, Brown.

“He was wait-listed at Harvard,” Ms. Lilly noted.

“He could have gone if he took off a year,” Ms. Bingham added, a tiny bit defensively.

The two women, as well as Mr. Bingham’s older sister Clara Bingham, were talking as if he was in the next room and might be eavesdropping on their conversation.

He wasn’t. Mr. Bingham, a writer of promise—who published stories in the New Yorker, as well as a novel, “Lightning on The Sun” and a collection of short stories, “Pure Slaughter Value”—died of a heroin overdose in 1999 at the age of 33.

His relatives had gathered at Ms. Lilly’s Upper East Side apartment late Monday afternoon to reminisce about him ahead of the 10th anniversary of the PEN/Robert W. Bingham Prize for Debut Fiction.

The award is given to a writer, much like Mr. Bingham himself, whose first work—a novel or collection of short stories—exhibits literary distinction but who could use some encouragement to complete a follow-up work of fiction. The award, which carries a $25,000 check, will be announced at the New School Auditorium on Monday evening.

“Every writer has a novel in them,” noted Clara Bingham, an author herself. “It’s the second act, with the pressure and expectations.”

It’s a pressure her brother, an heir to the Bingham newspaper-publishing fortune, apparently felt after the success of his short-story collection and while working on “Lightning on The Sun,” a novel that drew on his experience working in Phnom Penh for two years as a reporter for Cambodia Daily.

The book was published in May 2000, not long after his death. “It was agonizing,” his sister remembered. “He missed out on the glory. It was a beautifully written book and it got great reviews.”

His mother, an executive editor at Grove/Atlantic, added that “ Michiko Kakutani reviewed it in the Times. It was one of those reviews that tells the story and there weren’t any good pull quotes.”

One might assume it painful for his family to discuss a death as tragic as Mr. Bingham’s. “He wasn’t a regular junkie,” his sister explained. “He was an occasional user. And when mixed with alcohol it’s deadly. It was a total shock.”

But there seemed something therapeutic, even joyful about rousing his memory—the way he supported literary friends through “Open City,” a publication he joined in the early ’90s to showcase new writers; his all-round zest for life.

Mr. Bingham was born in Kentucky, where the family newspaper empire was centered. But his father, Worth, died in a freak accident when Robert was only three months old, and when he was two his mother moved the family to New York City and an attic apartment at the Dakota.

His writing, which bears the influence of Hemingway, Fitzgerald and Graham Greene—with perhaps a smattering of Jay McInerney and Whit Stillman thrown in—always seems to have one eye trained on the abyss.

“Tell about Dylan Thomas,” Clara Bingham instructed her mother.

“On Christmas Eve,” as a child, “Robert would always get up and recite ‘A Child’s Christmas in Wales,’” Joan Bingham said. “He’d learn a new stanza every year.”

But there was also the side of his personality that flirted with risk and danger. “There was a gonzo element to him,” his sister admitted. “He owned many BB guns.”

Ms. Lilly, an architectural historian who remarried and has two children with her second husband, John Lilly, recalled the excitement Robert felt the summer they rented a house in Rhinebeck, N.Y., and the owner encouraged him to shoot the pesky geese on his pond.

Indeed, his second New Yorker fiction piece, “The Target Audience,” is based on a true story involving a girl who got shot by a BB gun at Groton, where Robert attended boarding school.

“He was not the shooter,” attested Ms. Lilly who attended Groton at the same time her husband did.

This year’s candidates for the Bingham Prize were scheduled to read at the KGB bar in the East Village on Sunday night. But the annual event also doubles as an excuse for Robert’s old friends to gather.

“We get them all together,” Clara Bingham explained. “This gives us a structure to remember him.”

Write to Ralph Gardner Jr. at ralph.gardner@wsj.com