What exactly did this mechanic sabotage on the American Airlines 737, and how dangerous was it? An American Airlines mechanic sabotaged a 737 in Miami

An American Airlines mechanic accused of sabotaging a navigation system on a flight with 150 people aboard at Miami International Airport was denied bond by a federal judge on Wednesday after prosecutors suggested he may have links to a Middle East terrorist organization.
Abdul-Majeed Marouf Ahmed Alani, a 60-year-old veteran employee, told investigators after his arrest earlier this month that he disabled the system because he was upset over stalled union contract negotiations with the airline and wanted to generate some overtime for maintenance on the plane. He said he meant no harm to anyone, and the July 17 flight was aborted before takeoff after an error alert appeared on the navigation system.

His defense attorney said it wasn't as big a deal as they were making it out to be

But Alani’s assistant federal public defender, Christian Dunham, said the prosecutors were exaggerating the evidence. He pointed out that there was a second navigation system still working on the plane so his alleged sabotage could not have caused it to crash.
“We don’t believe he intentionally endangered the safety of people” on that flight, said Dunham, who sought a pretrial bond signed by Alani’s family members in California and Florida. “I think the government is blowing this out of proportion.”

What was it he sabotaged, and how serious would it have been if it had taken off?
It's a serious crime, but what that guy did, only affecting one system out of three, would be unlikely to cause a crash if the pilots were half way reasonably competent.  The take-off would be rejected as soon as the blocked side's speed tape failed to "come alive", well below 80kt (the pilot not flying checks that both speed tapes start to move as the airplane accelerates - this would be the first obvious clue and is a very basic action on a jet).