What could be ticking noise in 2003 cavilier air conditioner.?

could possibly be a lifter tapping (Ticking) ,depending on how many miles the vehicle has on it might be time for lifters to be adjusted,most foreign overhead cam engines must have lifters adjuster periodically ,if you turn on the air its puts more of a load on the engine causing a louder sound (also check oil level an engine low on oil will start to tick too.. the more wear the engine gets the looser the lifters can get over time ,look in your service manual if you have one and see what the intervals are on lifter adjustment this can give you an idea.
possible only if
the ticking sound is coming from the top of the engine usually comes from under the vavle covers where the lifters are located,

The Blend Vent Doors under the dash or the Fan Motor

Most likely its the relay on your air conditioner. Ticking sounds can be anything you really need a mechanic to listen. I say relay on air conditioner because is happened on my vechile an is still there.

doubtful the ac is causing the problem, I would check all your grounds, engine to body momentary lose of good ground can cause such a problem

Does it sound like the noise the turn signal makes? Does the noise change if you change the speed of the blower motor? Do all the function of the heater work (defrost, vent, floor, recirculate, fresh air, hot, cold)?

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This noise is caused by a defective AC/Heater air temperature control blend door actuator motor.

big end bearings( lower engine rod bearings to confuse) knock at idle and get louder with rpm
if the noise changes with rpm and you are hearing it near the ac compressor then it could be the ac pulley bearing
test if it is the compressor by listening and then turning the ac off the pulley will still spin but the compressor stops
if the noise remains , replace the pulley bearings

ticking can be a few problems that are not terminal first check oil if low top off and see if noise quiets down if oil good than have oil pressure checked -may be clogging or bad oil pump-check check where the ticking is coming from if its the head area than possible need to clean repair or replace a valve train part-[clogged push rod ,bad/need adjustment rocker sticking /bent valve oil return clogging /sludge] if coming from front of engine check the pulleys and water pump or fan for damage or debris a ticking noise usually is a noise that can be repaired just need to pinpoint the location

You may have something stuck in the intake.Piece of plastic bag maybe.Look on the pass. side .If you can't see the little square opening in the black plastic duct work you can put the blower on high ,set it to recirculate at mid-level and see if you can feel with your hand were the air is being sucked in.

Probably the vent door actuator......
Vent doors are controlled by small elect. Motors.