The important thing is remembering that what you want to measure is the time between pressing a button or joystick and the motor beginning to run. The best way to measure this is as follows (you need a digital oscilloscope):
- Open up a Logitech joystick and solder in two little wires to the a button's pads so that you get a current when it's pressed. Wire these to channel A of an oscilloscope.
- Attach a pair of leads to a motor controller. Wire these to channel B of an oscilloscope.
- Write software to run the motor when the button is pressed. Make it as simple as possible.
- Set the oscilloscope to trigger and acquire on channel A rise (or fall depending on the wiring). Then you can see the distance between the button signal's rising edge and the motor signal's rising edge.