2002 Oldsmobile Alero clicking sound

You more than likely have a faulty hazard lamp switch, quite comon with GM "N" body vehicles. A clicking noise or turn signals not working are the usual suspects when the switch is failing.
The switch is, last I knew, a dealer only item and was not too expensive and easy to replace.

The clicking sound from the hazard switch when the turn switch and hazard switch is in neutral is caused by dirty turn switch contacts in the muliti-fuction switch.

Take it out clean off the contacts and the problem goes away.

Do not buy a new hazard switch. The hazard switch is just responding to the inputs from the dirty turn signal switch.