The president said the museum pays tribute to "the true spirit of 9/11 — love, compassion, sacrifice."
"Like the great wall and bedrock that embrace us today," Obama said, referring to the way an underground flood wall withstood the attack, "nothing can ever break us. Nothing can change who we are as Americans."
One of the red bandannas Crowther made a habit of carrying is in the museum. Crowther's mother, Alison, said she hoped it would inspire visitors to help other people.
"This is the true legacy of Sept. 11," she said.
Former President George W. Bush was also invited, according to the museum. But Bush spokesman Freddy Ford said he was unable to attend because of a scheduling conflict.
At the dedication ceremony, retired Fire Department Lt. Mickey Cross described being trapped for hours in the wreckage — and then joining the recovery effort after being rescued.
Kayla Bergeron remembered taking her final steps to safety, after 68 flights, on the battered staircase that now sits in the museum. "Today, when I think about those stairs, what they represent to me is resiliency," she said.
Also in the collection are the shoes Florence Jones shed on her way down one of the towers.
"I wanted my nieces and my nephew and every person that asked what happened to see them and, maybe, understand a little bit better what I felt like to be us on that day," she said.
Associated Press writers Josh Lederman, Karen Matthews and Jennifer Peltz contributed to this report.