The correct answer is option 2: Watt.

Mechanical power is a measurement of the rate at which work is done or energy is transferred or converted in a mechanical system. The SI unit of mechanical power is the Watt, named after James Watt, the Scottish engineer who made significant contributions to the development of the steam engine.

Option 1: Joule is the SI unit of energy. While power is measured in Watts, energy is measured in Joules, which represents the amount of work done or energy transferred.

Option 3: Newton-second is the SI unit of momentum, not mechanical power. It represents the product of force and time.

Option 4: Joule-second is not a recognized SI unit and does not represent mechanical power. It may be confused with the SI unit of angular momentum, which is kilogram-meter squared per second (kg·m^2/s).

Therefore, the correct SI unit of mechanical power is the Watt, represented by option 2.