An illustrative example elevator governor includes at least one flyweight configured to move a first distance between an initial position corresponding to a zero speed condition and an activation position corresponding to an elevator speed that reaches a predefined threshold. A biasing member biases the at least one flyweight toward the initial position. The biasing member is configured to allow the at least one flyweight to reach the activation position when the elevator speed reaches the predefined threshold. A flyweight position member sets a rest position of the at least one flyweight in the zero speed condition that is between the initial position and the activation position. A range of motion of the at least one flyweight is limited to a second, shorter distance between the rest position and the activation position.