A few of the names were predictable. Ken Jennings, the “Jeopardy!” champion who holds the longest winning streak on the show, would be the ideal choice, some fans said.
Other suggestions were more unusual. What about Rosie Perez, the charismatic actress who, in “White Men Can’t Jump,” played a trivia geek who was determined to get on the game show?
As fans mourned the longtime “Jeopardy!” host Alex Trebek, who died at 80 on Sunday, they also began to speculate about who might replace him. Social media buzzed with suggestions and jokes about who should host the show, from mainstream options, like Mr. Jennings and the ABC News anchor George Stephanopoulos, to more jarring ones, like the brash comedian and podcast host Joe Rogan.
Mr. Trebek was not even the first host of “Jeopardy!” When the show was revived in 1984, some fans of its original host, Art Fleming, bristled at the cerebral Mr. Trebek, who was not afraid to show off his knowledge of trivia, said Claire McNear, whose book about “Jeopardy!,” “Answers in the Form of Questions,” was released on Tuesday.
Now, he is synonymous with the show.
“Trebek really felt like a member of the family because he was there in your living room every night,” Ms. McNear said.
It was understandable, then, that fans would ponder Mr. Trebek’s replacement even as they grieved him, she said.
So, at least according to the fans, who are the contenders?
Many fans on social media were pushing for LeVar Burton, who hosted “Reading Rainbow”; played Lt. Geordi La Forge, the brainy, blind engineer on “Star Trek: The Next Generation”; and portrayed Kunta Kinte in the 1977 mini-series “Roots.”
By Tuesday, more than 1,500 people had signed a petition urging Sony Pictures Entertainment Inc. and the show’s executive producer, Harry Friedman, to give Mr. Burton the job.
“LeVar Burton has inspired and shaped the minds of several generations of trivia-loving nerds,” the petition said.
Mr. Burton has long been a fan favorite to replace Mr. Trebek, Ms. McNear said. And before Mr. Trebek’s death, Mr. Burton was not shy about his interest in taking on the role.
“Not gonna lie, I feel like I’ve been preparing my whole life to occupy the @Jeopardy host podium when Alex retires,” he wrote on Twitter in September.
Mr. Trebek had said he would like to see a woman or a person of color succeed him, according to Ms. McNear, who interviewed him for her book.
“He did say to me that he was keenly aware that the vast majority of hosts were white men,” she said. “And it would be wonderful if the next host of ‘Jeopardy!’ did not look like him.”
Mr. Jennings is another fan favorite. Not only did he win a record 74 consecutive games and $2.52 million on “Jeopardy!” in 2004, he is also a consultant on the show.
His serious yet genial demeanor would make him a natural successor to Mr. Trebek, said Steve Macek, a professor of communication at North Central College in Naperville, Ill.
“Fans would be upset if Trebek were replaced by some colorful or funny personality who upstaged the contestants and the questions,” he said. “Ken Jennings, the winningest Jeopardy contestant, has to be in contention, and he’d be a good choice.”
And Mr. Jennings wants the job, or at least he did two years ago when Ms. McNear interviewed him for her book. “He told me that he would absolutely serve his country and serve his game show if he were asked,” she said.
There was also fan interest in Ben Mankiewicz, the mild-mannered Turner Classic Movies host, even if a petition urging the producers of “Jeopardy!” to make him the host had not amassed nearly as many signatures as the one in support of Mr. Burton.
“Ben is someone who can grow with the current and future viewers for many years to come, in the way Alex did,” wrote M.J. Rocissono, the fan who started the petition.
“I can think of only one appropriate host right now and that is @BenMank77,” said one fan on Twitter, using Mr. Mankiewicz’s handle.
Ms. Perez’s turn as a “Jeopardy!” obsessive in “White Men Can’t Jump” (1992) remained memorable enough to spark support from several fans.
“If the new Jeopardy host isn’t Rosie Perez then I don’t want it,” one fan wrote on Twitter. “Rosie or Bust,” another said.
Not many other women have been cited by fans as a possible host, even though Mr. Trebek championed the idea of a woman taking over. In 2018, he said he thought Laura Coates, a CNN senior legal analyst, would make a good host.
Still, Sarah Kornfield, a fan of the show and a professor of communication at Hope College in Michigan, said it was unlikely that the producers would choose a woman to fill the role. “Expertise and warmth and trustworthiness are associated with masculine characteristics in the United States,” Professor Kornfield said. “People associate women experts with being shrill, and that is not the tone that ‘Jeopardy!’ is supposed to set for the night.”
She said she was personally rooting for Melissa Harris-Perry, an American politics professor and former MSNBC host.
Mr. Rogan, the popular but divisive podcaster, was on the minds of some fans who said his unpolished persona would give the sober-minded quiz show a much-needed jolt.
“Joe Rogan needs to host Jeopardy,” one fan wrote. “He would have a field day making fun of the contestants.”
But Mr. Rogan, who presents himself as a counterweight to what he sees as political correctness run amok, has drawn criticism for discussing conspiracy theories, belittling transgender people and sharing his platform with far-right figures, such as Gavin McInnes, the founder of the Proud Boys.
Ms. McNear said he should not be considered a serious candidate to succeed Mr. Trebek.
“‘Jeopardy!’ fans would be rioting in the street,” she said. “The institution must be protected.”