Mrs. Trump’s staff plans to continue to solicit ideas for the initiative, which focuses on opioid abuse, social media pressures and mental health issues among young people, from places the first lady has visited. Those include a West Virginia clinic that treats infants born with opioid addiction and a Michigan school where students participate in a program to stress the importance of emotional intelligence and kindness.

But critics did not buy the defense. Mrs. Trump enjoys higher popularity ratings than her husband, but she has been accused of plagiarism before. In 2016, a large part of a speech she delivered at the Republican convention appeared to be taken from remarks Michelle Obama, her immediate predecessor, delivered in 2008. On Monday, observers also noted that Mrs. Obama had delivered remarks in 2016 urging men to “be better.”

Like her husband, Mrs. Trump is also viewed by many as having baggage related to the Obamas, stemming back to jabs she took at President Barack Obama while her husband was promoting the conspiracy theory that Mr. Obama was not an American citizen: “It’s not only Donald who wants to see” Mr. Obama’s birth certificate, Mrs. Trump said during a 2011 interview. “It’s American people who voted for him and who didn’t vote for him.” Mr. Obama eventually released his long-form birth certificate.

The East Wing is known to lash out when faced with coverage the first lady views as unfair: In October, Mrs. Trump directed her office to release a scorching statement criticizing President Trump’s first wife, Ivana, for jokingly referring to herself as the first lady. But East Wing aides who had worked to execute a tightly scripted and controlled kickoff — and who are particularly sensitive when observers draw comparisons to the Obamas — had not anticipated the negative reaction to the booklet.