Analysis by Michael Warren, CNN

(CNN) -- Freshman Democratic Rep. Ilhan Omar got in elevator on Capitol Hill Tuesday as CNN's Sunlen Serfaty shouted a final question: Did the Minnesota Democrat believe President Trump was a white nationalist?

Without speaking, the congresswoman arched her eyebrows, as if to say, "What do you think?" as the doors closed.

After a stormy start to her term in the House, marked by the controversy over her comments about the influence pro-Israel donors have over some lawmakers, Omar had kept a low-profile for the past month. But the 37-year-old Somali immigrant, one of the first two Muslim women to serve in Congress, re-emerged this week with a tweet calling one of Trump's top White House aides, Stephen Miller, a "white nationalist."

"The fact that he still has influence on policy and political appointments is an outrage," she wrote.

Outrage has a way of following Omar, who received a death threat last month at her Capitol Hill office. Her latest tweet prompted another round of outrage as Republicans denounced her for saying Miller, who is Jewish, is a white nationalist. The President's son Donald Trump Jr. tweeted Monday night that Omar had "no shame" and called her the "head of the Farrakhan Fan Club."

One GOP strategist took to Fox Business Tuesday to accuse Omar of "assaulting Jews." To top it off, President Donald Trump tweeted a video clip of the segment.

Omar showed no signs of backing off, responding to Trump's tweet by quoting her 6-year-old daughter: "Knock it off. You're the president."

The episode demonstrated a new reality for Washington: In a few months, Omar has displaced Nancy Pelosi as the Republicans' favorite liberal boogeyman, a new face the GOP can weaponize in an attempt to depict the entire Democratic party as extreme and out-of-touch.

This puts Democrats in a difficult position: either defend their colleague against Republican attacks or to keep at arm's length a controversial member who engages in over-the-top rhetoric many Democrats consider problematic or even bigoted. For her part, Omar seems to be at once surprised at becoming a polarizing figure while also embracing the role.

Appearing on "The Late Show with Stephen Colbert" on Wednesday night, Omar made her most fulsome comments about the issue, responding to Colbert's question about what it's like to have been used as a political cudgel during her first few months in Congress. "This whole process really has been one of growth for me," Omar said.

She then referenced a conversation she had recently with her predecessor, former Minnesota Rep. Keith Ellison, who is Muslim and now serves as the attorney general of Minnesota. Omar said they were discussing things that historically have been offensive to African Americans, some of which she told Ellison she did not find offensive. Ellison, in turn, told her that is because she is an "African immigrant," Omar said, and that "none of those things hold history for you."

The point, Omar seemed to be making, is that she perhaps lacked a clear understanding of the full history of anti-Semitic views in the US.

"Oftentimes when you are speaking you might not understand the historical context of some of the words that you might use and the kind of pain it might incite for people," Omar told Colbert. "And so, in this process I am learning that everything is not as simple as we think we might want to state it."

The GOP's Enemy No. 1

While Republicans have trained fire on other Democratic freshmen congresswomen, including Alexandria Ocasio-Cortez and Rashida Tlaib, they seem to have settled on Omar as their most potent weapon to use against Democrats.

Last week, a political organization with ties to the GOP establishment launched a digital ad campaign highlighting those comments and urging Democrats to remove her from the Foreign Affairs Committee. The American Action Network, a nonprofit, does not disclose its donors, but its sister PAC, the Congressional Leadership Fund, includes among its top donors casino mogul Sheldon Adelson and his wife Miriam. The Adelsons have long donated to both the Republican Party and to pro-Israel political advocacy groups.

The 30-second ads, backed by AAN's six-figure buy, are running online in at least four House districts held by freshmen Democrats. They charge that Omar "keeps hurling anti-Semitic slurs" and includes images of a now-deleted tweet from 2012 in which she stated "Israel has hypnotized the world."

The official Republican Party is focusing on Omar as well. On Monday, a spokeswoman for the National Republican Congressional Committee called Omar and Tlaib "bigots," and linked the two members to a newly announced congressional candidate in Michigan.

"Inspired by bigots like Ilhan Omar and Rashida Tlaib, socialist Jon Hoadley announced he will run for Congress in 2020," reads an NRCC blog post by spokeswoman Carly Atchison. Hoadley, who is running against longtime Republican incumbent Fred Upton in Michigan, worked last year alongside Omar on a progressive think tank project called America's Cabinet.

And on Tuesday, the Republican National Committee's rapid response team blasted out an article from the right-wing Daily Caller highlighting a recent speech by Omar at a fundraiser for the Council on American-Islamic Relations. "CAIR was founded after 9/11 because they recognized that some people did something, and that all of us were starting to lose access to our civil liberties," Omar said on March 23. (In fact, CAIR was founded in 1994.)

"Ilhan Omar downplays terrorism AGAIN," read the RNC's email.

Omar has even gotten on the radar of state-level Republicans. At a March event sponsored by the state GOP at the West Virginia statehouse, someone displayed a poster associating Omar with the Sept. 11 attacks. The West Virginia Republican party denounced the poster; an official at the state capitol who defended it ended up resigning her position.

It's likely too early to determine whether the focus on Omar will benefit Republicans at the ballot box. Unlike Pelosi, who for years has been a nationally-known political figure, Omar is relatively new to the national scene. It's unclear what effect she will have on voters, whether she'll swing them away from Democrats or motivate Republican base voters.

No national polling firms have asked voters about Omar or her recent controversial remarks about Israel. Support for Israel over the Palestinians among Americans has generally hovered around 60% for more than a decade, according to Gallup.

A dilemma for Democrats

What the GOP scrutiny on Omar does accomplish, however, is putting her fellow Democrats in a bind. Moderate Democrats have been forced to respond to her comments, and the results have been awkward. Democratic New York Rep. Tom Suozzi, in an on-air interview Tuesday with CNN's Brianna Keilar, struggled to avoid addressing Omar's tweet about Miller before admitting that her type of rhetoric "makes it more difficult" to reach solutions on immigration.

"We need to take the temperature down," Suozzi pleaded.

Efforts by House leaders such as Pelosi to curb Omar's taste for the polemic have only worked to temporarily tame her Twitter feed. A House resolution in March denouncing anti-Semitism spearheaded by pro-Israel Democrats received pushback from progressive allies of Omar. Why, they asked, was their colleague being singled out when Republicans like Iowa Rep. Steve King and even the President had said worse things about immigrants?

Under pressure, the language of the resolution was changed to include condemnation of all hate. Jewish and pro-Israel Democrats became upset the new resolution downplayed the seriousness of Omar's comments, and moderate Democrats wondered why the caucus had been diverted from addressing the pocketbook issues they had run on.

Michael Meehan, a Democratic strategist, called the scrutiny on Omar by Republicans "rubbish politics" and said he was skeptical it would hurt Democrats. "Clearly they're looking for another boogeyman," Meehan said of Republicans.

"Permanent residency"

If Omar intends to stop speaking out, either to deny Republicans the ammunition or to acquiesce to Democratic leadership's wishes, she shows no signs of doing so. She sees her notoriety as a political asset, burnishing her credentials as a fearless speaker of truth to power.

It's a role she seems to relish. After Vice President Mike Pence and Democratic majority leader Steny Hoyer made oblique criticisms of Omar in their speeches before the American Israel Public Affairs Committee conference late last month, CNN asked for comment.

"It's been interesting to see such a powerful conference of people be so fearful of a freshman member of Congress," Omar told CNN on March 26. "I hope that they figure out a way to not allow me to have a permanent residency in their heads."