Still, some family members who sued would like to see the remains encased in a kind of "tomb of the unknown soldier."
A key new element in the case is Superstorm Sandy, which flooded the World Trade Center site in October, filling the underground space with water.
If the remains were at the site when the storm hit, "body parts would be floating all over Manhattan," said Jim Riches, who lost his 29-year-old firefighter son, Jimmy Riches, in the terror attack.
"They promised us the remains would be separate from the museum," he said. "But they're going to be in the basement of the museum."
Riches and others present for the appeal spoke on a Madison Avenue sidewalk opposite the courthouse, their anguished voices rising above the midday Manhattan traffic. Some wept, clutching photos of their dead.
Rosaleen Tallon, sister of fallen firefighter Sean Tallon, held up a photo of the trade center site on the night the storm hit, sending water gushing over the concrete under construction.
"This picture shows us water pouring into ground zero," she said. "And we feel like nobody cares about the remains of our loved ones anymore."