In some Republican circles, the notion of President Trump’s children running for political office is not only a parlor game — it’s a matter of finding the right opportunity for the right Trump.

Could that opportunity exist in blue New York? And could it happen now?

Some state Republicans are eager to give it a try. With Representative Peter King, a Republican, not seeking re-election on Long Island next year, the circumstance may never be better for a younger Trump to run.

But which one?

Donald Trump Jr., the president’s elder son and the member of the family who is most naturally fluent in the language of the Republican base, has frequently been mentioned as a possible candidate for office.

He attracts a younger contingent of Republicans to events, organizers say, and has demonstrated his ability to raise big money for congressional candidates on Long Island.

Lara Trump, who is married to the president’s son Eric, is a senior adviser on the president’s re-election campaign; her potential for future office has also been mentioned by some close to the family.

“Among Republican voters, Trump is still a magic name,” Mr. King said in a telephone interview. “I’ve been with Don Jr. several times on Long Island. Was with Lara once at a fund-raising event. She handles herself very well. She’s good with people.”

The buzz around a potential junior Trump candidacy heightened last week, when a poll surfaced in New York that showed Lara Trump winning by more than 30 points in a hypothetical Republican primary for Mr. King’s seat.

The poll, conducted this month and paid for by an influential conservative anti-tax group, the Club for Growth, was picked up by the right-wing site, Breitbart, and quickly gained steam.

David McIntosh, the president of Club for Growth, acknowledged that Ms. Trump had nothing to do with the poll; he said the committee’s intent was to show Ms. Trump how popular she was in order to lure her into the open race.

Ms. Trump hails from North Carolina and now lives in Manhattan. She is not currently a candidate on Long Island, or anywhere else.

“We often do that to test people who we think would be a good candidate,” Mr. McIntosh said of including her in the survey, which the group said cost about $3,000.

Ms. Trump declared herself “incredibly honored” by the results of the poll but uninterested in running.

For now.

“While I would never close the door on anything in the future, right now I am focused on winning a second term for President Trump,” Ms. Trump said in a statement.

The poll might have been a lark. Or perhaps it was a way for Club for Growth, staunchly opposed to Mr. Trump in the 2016 election, to ingratiate itself to the president and his supporters. (The group is backing his re-election.)

Some local Republicans saw it as a bit of political score-settling in the way it showed Ms. Trump soundly besting the only other named candidate, Rick Lazio, a former Long Island congressman. Mr. McIntosh called the moderate Mr. Lazio a “repeat loser,” but added: “I like Rick Lazio. There’s no personal animosity there.”

Asked about the poll, Mr. Lazio said in an email that “to the best of my knowledge I have never met Lara Trump,” and added that he had “no knowledge as to whether she has ever lived in, been employed in and has had a meaningful involvement with the people of the Second Congressional District.”

Whatever the reason for the survey, it underscored the popularity of the Trumps, particularly on Long Island.

The children are popular among party activists in New York, so much so that Republican fund-raisers practically swoon over the idea of getting one of them to headline an event. Donald Trump Jr., has been particularly prolific and lucrative, making several trips this year to Long Island.

“They’ve been extremely helpful to the local party,” Mr. LaValle said of the president’s family. “They’re certainly a known commodity here.”

The event, at a St. James, N.Y., catering hall, drew more than 350 people and raised more than $200,000 for Mr. Zeldin’s re-election, according to his campaign.

Mr. Trump has repeatedly been floated by supporters as a potential future candidate for president, and, much less realistically, for New York City mayor, a tall order given the city’s strong Democratic base and its intense dislike for the president.

“Listen, I don’t ever rule anything out,” Mr. Trump said on “CBS This Morning” this month when asked about his political future. He declined to comment for this article.

A year earlier, he told The New York Times that he loved “the intensity of campaigning,” but admitted that he was less sure how much he “would love aspects of the actual job.”

The fund-raiser on Thursday appeared to be at least the third time that Mr. Trump helped raise money for Mr. Zeldin, who has risen to party prominence as a vocal defender of the president and his family.

“Nobody, other than the president himself, brings more star power, excitement and energy than members of the Trump family,” Nick Langworthy, the state Republican Party chairman, said in a statement. “It’s a special phenomenon that no other president has enjoyed, outside of maybe the Kennedys.”

At the same time, the political window for Lara Trump or any other Trump relative to run in a place like Long Island may be closing. Democrats now outnumber Republicans among registered voters, making gains in both Nassau and Suffolk Counties.

And the Trump name is almost equally galvanizing among Democrats as it is magnetic among Republicans, and a local race involving one of them could draw big money from activist donors from both parties.

“There’s real solid support and real solid opposition” to the president, Mr. King said of his district. “A tremendous amount of anti-Trump energy.”

Ms. Trump could win, he said. “At the very least it would be very competitive.”