Your Saturn has a factory security system called "Passlock". There is a signal generator built into the ignition switch lock housing; when the correct key is turned in the ignition this sends a signal to the car's body control module. The BCM compares this signal to the one programmed into it's memory; if they match the BCM then sends a message out over the car's data network that is received by the powertrain control module; once this message is received the PCM then allows the fuel injectors to operate so the engine will start.
To find out where the problem might be in your car, you will need to take it in to a GM dealer or a repair shop that has a Tech-2 scan tool. The security system can generate fault codes just like the other electronic systems in the car, but the ONLY scan tool that can access the security system to see what is going on is the factory one (tech-2).
The most common failure in this system is a faulty ignition lock housing (around $250). However, an internal BCM fault or a poor electrical connection somewhere in the system can have the same effect.
This is not really the kind of problem you can accurately diagnose at home, since the factory scan tool is needed for testing' for that reason you may want to take this one in to your mechanic to be fixed. If not repaired, eventually the trick of leaving the ignition on for 15 minutes will not work consistently and you may end up having to be towed in (which can often cost more than the repair).
Yes, you can replace that part yourself, if you wish. After replacing it you will need to do a key relearn by turning the key to run, allowing it to sit for 10 mins (like you have been doing), then cycle the key off and back on, wait another 10 mins untill the light goes out, and repeat the key cycle and wait one more time. After the light goes out the third time the engine should start if the relearn was successful (a 30 minute process total).
I am afraid I have no current part number for this component; when Saturn went out of business a lot of their part numbers stopped appearing in the parts guides. Your local Gm dealership may be able to obtain this component for you though if you give them a call. Replacing it takes an experienced GM mechanic about 2 1/2 hours; if you are not familiar with this type of repair you may want to double that.
First, i would suggest having the problem diagnosed by a shop with a TEch-2; replacing a $250+ part that takes several hours to change on a guess and not having it fix the problem would be frustrating to say the least. $100 or so in diagnostic labor may be a good investment before you at tempt to repair anything.
Thanks for letting me know the outcome; however the fix is indeed puzzling since the key has no effect on this type of security system other than mechanically releasing the wafers in the ignition lock. All of the security system components are external to the key on this model.
The costs being quoted to you by the dealership do seem quite higher than normal; if you ever do need a shop with a tech 2 you might want to call around to see if any of the independent shops in your area have this equipment; many do. Lock cylinder replacements generally are in the $250 range and BCM replacement and programming is usually under $600 in most areas.
You certainly can't argue with results though; let's hope it stays fixed and again, thanks for the update!