I have been called upon to assist a neighbor lady in fixing a problem she is having with her furnace. It is the same as my own, a 28 y/o Carrier (model #396GAW000075). It starts its cycle without any problems. After it has been running for about 10 minutes or so, the pilot ignitor starts to click/spark. The clicking gradually picks up speed and then there is a single louder click as if the pilot is trying to light (this is just a guess and I may be totally wrong as to what is happening at this point). This continues until the furnace turns off. I would greatly appreciate it if someone could diagnose this problem and how do I go about fixing it. I am trying to help my neighbor save a few dollars.

Take the cover off the burner compartment of the furnace. Observe the sequence of events that occurs at the furnace when you turn up the thermostat and post that sequence of events in order and in detail.

You aren't observing what is happening in enough detail to be able to determine what the problem is.