Not going to be the brake switch you're hearing, unless you're stepping on the brake at the same time.
Probably broken wiring in your multifunction lever as described earlier, and the click you're hearing is the cruise kicking out as you let go of the button. You can test it with a multimeter pretty easily. Under the dash near the ALDL (diag connector) will be a harness socket clipped to the dash support, with the cruise wiring plugged into it. Kind of easy to find as it's a flat cable with a shiny black sheath, sort of looks like a telephone cord (flat and straight, not curly) and has a bright orange connector on the end.
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Has four connections on it. One is power from the fuse block (pink/black) the other three are the cruise functions. On/off, set/decel, resume/accel. Should be able to look at the harness socket to see which 'finger' on the connector is the power wire. Been a while since I've had to test one so I can't remember. Anyway you can't tell looking directly at the connector because all the wiring is covered in the black jacket. Anyway meter set on continuity...unplug the connector, put one meter probe on the finger for power, then other probe on the next finger and try each of the three switches. You should be able to get a beep when the switch for that finger is closed. Move to the next finger, etc. So you can see which of the three switches aren't working or are flaky (might help to have someone gently jiggle the lever while trying the switches, to see if it's intermittent.)
Not too hard at all. The lever pulls out of the column by gently tugging it straight out. It's held into the column by the actual electrical portion of the wiper switch, and has a sort of check ball/spring type retainer that takes a small amount of force to remove. Don't tug it too hard as you don't want to rip it away from the wiring. If the wiring breaks off inside the column you're screwed. At the connector under the dash, unplug it, and run a several foot long piece of mechanic's wire through the hole, bend it over so it can't pull out. Then gently pull the cruise wiring up through the column from where the lever was removed, pulling the mechanic's wire along with it. Then use the mechanic's wire to pull the new cruise wiring back down through. TAKE YOUR TIME...if you manhandle this job you'll regret it.
Richard