WASHINGTON — A central issue in last week’s impeachment trial was whether former President Donald J. Trump deserves a political future. But his acquittal sparked speculation on Sunday about the electoral prospects of another Trump: his daughter-in-law, Lara.

Senator Richard M. Burr’s decision to vote for the conviction of Mr. Trump incensed many Republicans in his home state of North Carolina, and in doing so reignited talk that Ms. Trump, a native of Wilmington, N.C., would seek the Senate seat Mr. Burr will vacate in 2022.

“My friend Richard Burr just made Lara Trump almost the certain nominee for the Senate seat in North Carolina to replace him if she runs,” Senator Lindsey Graham, Republican of South Carolina, said in an interview on Fox News on Sunday.

Ms. Trump did not respond to a request for comment. One senior Republican official with knowledge of her plans said that the Jan. 6 riot soured her desire to seek office, but that she would decide over the next few months whether to run as part of a coordinated Trump family comeback.

If negotiating a post-Donald-Trump world has been a disorienting experience for Republicans around the country, it is especially acute in North Carolina, a state that has become a polarized, and nearly deadlocked, partisan battleground.

Mr. Burr’s vote, and the torrent of criticism among North Carolina Republicans that came with it, appeared likely to sharpen the differences in the primary to succeed him between staunch Trump loyalists and Republicans who see a need to appeal to educated suburban voters in a state with steadily changing demographics.

“The G.O.P. base is getting smaller,” said Paul Shumaker, a veteran party strategist in Raleigh.

It was not just Mr. Burr’s vote that inflamed the party’s rank and file. While the state’s junior senator, Thom Tillis, who was re-elected last year, voted to acquit the former president, Mr. Tillis used his statement after the vote to all but invite prosecutors to indict Mr. Trump, saying the former president’s “ultimate accountability is through our criminal justice system.”

Mr. Trump’s allies predict that such talk would prompt a revolt from the right that would result in the election of more pro-Trump candidates. And, the thinking goes, who could be more pro-Trump than an actual Trump?

Ms. Trump, 38, a former personal trainer and television producer who graduated from Emsley A. Laney High School in Wilmington and from North Carolina State University in Raleigh, has been floating herself as a possible Burr successor for months.

Another Republican, former Representative Mark Walker, a Trump ally, has already announced his candidacy, and Pat McCrory, a Republican former governor, is considering one. Mark Meadows, the former North Carolina representative and former Trump chief of staff, is also said to be in the mix.

“We are going to take a very long look at all the candidates versus, you know, some kind of coronation,” said Mark Brody, a member of the Republican National Committee from Union County, outside Charlotte.

Doug Heye, a former Republican National Committee spokesman who used to work for Mr. Burr, questioned whether Ms. Trump was willing to endure the tussle and tedium of running or serving. “Many people love the speculation and the attention, but being senator is a lot of hard work,” he said.

First, however, there is the question of her residence. Ms. Trump currently lives with her husband, Eric, and their children in the northern suburbs of New York City and would have to move back.

Then there is the less straightforward question of branding. The Trump family name is a wild card — it will be a plus with loyalists and fund-raising nationally, but it could be a liability in a battleground that the former president won by a mere 1.3 percentage points in 2020. There is also a possibility Ms. Trump’s candidacy could help increase Democratic turnout, especially among the state’s large Black population.

Or it might be a wash.

“There is a myth that Trump voters will come out for Trump candidates or family members,” said John Anzalone, a Democratic pollster and a veteran of campaigns in the South. “Cult members only come out in full force for the cult leader.”

That Ms. Trump’s may-or-may-not-happen candidacy is generating buzz is, in itself, a reflection of the party’s anxiety over its future.

Ms. Trump’s boosters, led by Mr. Graham, view her presence as a way to weaponize the backlash against Mr. Burr’s vote, seen as a betrayal sufficient to warrant a rebuke by the North Carolina G.O.P. over his “shocking and disappointing” decision.

Others simply see Ms. Trump as a potentially well-funded candidate with the built-in advantage of sky-high name recognition.

Representative Patrick McHenry, a Republican who represents the Greensboro area, downplayed the importance of Mr. Burr’s vote but said Ms. Trump would “be the odds-on favorite” if she runs.

“No one comes close,” he said.

Glenn Thrush contributed reporting.