10,000 feet wasn't an entirely arbitrary choice for the sterile cockpit rule. That altitude (as noted above, pressure altitude above MSL) corresponds with other FAR requirements, such as the requirement for use of supplemental oxygen in unpressurized aircraft (for periods >30 minutes above 10K - goes to full time use above 12K), airspeed restrictions, and others.

There is a general logic as well. Generally, it's more likely that an aircraft, especially a commercial airliner, is engaged in more critical flight operations at lower altitudes as opposed to cruise flight. Obviously the closer an aircraft is to the ground the less margin for error and recovery. And below 10,000 MSL is where the majority of general aviation occurs, much of which is under visual flight rules - hence there is more traffic that isn't under positive control which pilots must be more keen to see and avoid.

Could the FAA have chosen 9,000 feet or 11,000 feet? Sure - freeway speed limits could be 68 or 73 MPH, too, but at some point you have to pick something reasonable and go with it. MSL altitude is used because that's what pilots are using as a reference to operate the aircraft at those altitudes. Yes, it does result in the height above ground level being lower above Denver than above Miami, but presumably it was determined that 10K MSL was still sufficient across US airports.