WASHINGTON — Lisa Monaco was President Barack Obama’s top counterterrorism adviser when she was handed an intractable problem: Fix the administration’s ineffective response to the kidnappings of Americans by Islamic State fighters, which had prompted outcries from victims’ families, without changing the government’s refusal to make concessions to terrorists.

Ms. Monaco quickly instituted a change, according to Matthew Olsen, a former director of the National Counterterrorism Center. She mandated that the families, who had been kept in the dark about the government’s restrictions and had even faced threats of prosecution should they pay ransoms themselves, be brought into the fold. Most had lost faith in the government, and she sought them out to ensure that a new hostage policy was fair and credible.

“For the administration to realize it was not handling this right was a lot to Lisa’s credit,” said Diane Foley, whose son James Foley was the first American to be beheaded by the Islamic State in 2014. After Ms. Monaco’s team completed its review, the administration adopted a policy that included advising families of all their options and refraining from threats of prosecution. Mr. Obama acknowledged that the government should have treated them as “trusted partners.”

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