I've found only a few postings on the net about intermittent starting problems for a C5. Description: I place the key in the ignition, foot on the clutch, other foot on the brake, radio off, AC/Fan off, turn the key and nothing happens except for the gauges registering to the right. Everything appears completely normal except the car won't start. Battery is good. In fact, the battery is new. I receive no messages on my DIC. This is happening about 50% of the time or more. (It's a lot of fun when I have a date in the car too.) How do I get it started you may ask... It will always push start but can't always do that, however that at least rules out the pass-key system, right? Otherwise the security would not permit push starting. Well when this intermittent starting happens, I turn the key multiple times, sit there, take the key out, wait a mintue, swear..etc.. and it randomly starts up when I least expect it. It's even starting up when the key has been slowly turned back 1/32" to the left. It's started when I've pumped the clutch...It's started when I've just sat there doing nothing but have the key to the right. It's completely random. Vibration or stillness doesn't seem to matter.

I've done as much as possible that I know of. Here goes.
I've tested the starter. (I talked to a starter mechanic to understand how they work and he indicated that the contacts could be remaining in an open position therefor preventing any turning or sounds whatsoever....so I had it tested.)
I've tested and essentially dismantled (twice) the Clutch Pedal Switch found under the clutch pedal arm.
I've had the ignition cylinder replaced.
I've had the keys tested at the GM dealer and apparently they're "tight.".
I've had an "interrupt relay" replaced. (I'm not sure if this is the same as the starter relay.
I've had the steering column lock replaced. I've even temporarily installed a CLB for lack of any other ideas!
I've pulled the fuse box in the passenger side and checked every single fuse.
I've put a brand new battery in it and checked the cables.

Description of what happens:

Put the key in the ignition. Turn it to the right. All the gauges register, fuel pump is heard (if the window is down), then nothing. I do here the relay click if I pump the clutch pedal. The switch at the clutch pedal triggers the relay on the passenger side so I can tell that's connecting. I don't get any messages on the DIC. I don't get any turning or any other sounds. It doesn't seem to matter if the air is on or off, radio on or off. I turn them all off anyway. I'll sit there and play with the key. Back and forth, in and out and even just take it out and wait a minute or two. It usually starts at some random moment. Keep in mind this has happened now for the past two years and only recently appeared to be worse. It has happened to the extent where I've not been able to start it so I've actually been in a few situations where I've had to have it push started. That's another reason I don't think it's related to the Pass-Key system and anti-theft mechanisms because then the car wouldn't be allowed to start. I'm at a loss. I've had it in to a mechanic that works only on corvettes three or four times and it's started every single time for him. Go figure. Any help whatsover is greatly appreciated.

 

Sorry to hear of you problems. You sound like you've tried all the "normal" things that would be associated with something like this.
It appears to me that you may have one of the worst possible problems to try track down and that would be the dreaded intermittent open circuit. It's the inconsistency of the problem that will drive you up a wall.
Question: Have you (or anyone else) made any changes/additions to anything on the car related to the electrical system? Leaving the CLB that you mentioned aside, is there anything else? Kill switch? Stereo work? Hardwired radar detector? Sometimes other people's "work" can cause you problems down the road.
Have you determined if you are getting current flow to the starter solenoid itself while the key is in the "start" position? If not you just may have to start working your way backwards so-to-speak, in the circuit to see where the required voltage is present and where it isn't.
Another thing to try would be to remove the driver's side knee bolster so you can gain access to some of the cable harnesses and while holding the key in the start position, move some of the cable bundles around a bit and aee if the starter all of a sudden cranks. This may be a bit of a pain since the clutch switch has to be closed in order for the starter circuit to be complete, but you can bypass the switch with a temporary wire just for the purposes of testing.
Don't rule out loose connections at or on the starter itself too.The DIC and gauges don't require much current to operate but the starter can draw upwards of 150 amps on many vehicles. You might even try the "turn and hold the key" technique while you or someone else does some moving around of the cable harnesses underneath. Check the engine block grounds too . Naturally you have to be prepared for a sudden starter engagement, so be careful. HTH
BTW, the relay you mentioned replacing is the TDR (Theft Deterrent Relay)