"What we want is justice and to give a family the relief and honor they're entitled to by having this man's name on the wall," Weisenberg said.
Brady has been nominated to be included on the memorial in Albany that lists 2,366 names of firefighters killed in the line of duty since 1811. The selection committee, however, has voted against it four times. The memorial's criteria for inclusion defines "in the line of duty" as activities related to emergency incidents or training.
Brady, 42, was working on top of a Malverne Fire Department truck in the firehouse when a colleague drove it out without realizing he was there. Brady was crushed between the vehicle and a ceiling beam.
His death was determined to be in the line of duty by the state Workers' Compensation Board, U.S. Department of Justice and National Fallen Firefighters Foundation.
Weisenberg's bill would circumvent the memorial committee, allowing the recognition by other authorities to count.
"Everywhere else he's accepted as dying in the line of duty," Weisenberg said. "Why should he not be on the wall in New York State?"
Last year, Newsday reported that professional firefighters on the committee have voted against including Brady while volunteers have voted in favor.
Thomas LaBelle, executive director of the New York State Association of Fire Chiefs, said some have argued that to be on the wall a firefighter's death should have been heroic, not merely tragic.
"Tragedy is enough," LaBelle said. "If it's in the line of duty of serving your community, the name should go on the wall."
Cuomo's office declined to comment. The governor has until the middle of next week to sign or veto the bill.