2001 Jeep Cherokee while braking.

The way I found this the last time was by running it in gear on a hoist. The sound could be replicated with the wheels off. It was found to be a spot of rust on the backside of the rotor the do-it-yourselfer didn't clean off that was causing it to wobble, (just as a warped rotor would do), and the caliper was sliding sideways back and forth twice per wheel revolution. There were two deep grooves worn into the knuckle that the pads rode on, and the metal plates of the pads were catching on those grooves causing the clicking noise. The formation of those grooves can be reduced by using high-temperature brake grease on those contact points.