94 Suburban that has an inside tick

Hello. I am the proud owner of a 94 suburban. I rarely drive it (35,000 miles), but when I do, it provides humor for my two young daughters. As soon as I start the car, it begins making this tick/clicking noise right at the passenger side foot-well. It only lasts for about 30 seconds and goes away. It only comes back when driving, if I turn the A/C or heat on (again for 30 seconds). The A/C and heat work fine. The fun is when I turn off burby (as my youngest has named it), it tick/clicks for another 30 seconds with the key out of the ignition! Oh yeah, it also leaves a smell of oil on our clothes?not too bad, just enough to make no one want to ride with me. I welcome any suggestions and thanks in advance for your thoughts.