Based on the symptom, I'd have to say the CDI box that fires the coil is shot. It either isn't building up enough voltage, or storing enough voltage at the coil, to get more than the one zap. First thing I'd swap out. The CDI sits up, lets some current through, good spark, then it is taking what, minutes/hours/days before it allows the coil to resaturate and fire?
-WD

Good responses so far.....

WD - that was partially my thought too.  One shot from the CDI.  He assures me that he checked the contacts on the CDI, too.

Slavy - in checking things, I got him to reopen the case and check the pickup.  What he did find was cracked insulation on the pickup leads where they passed through the grommet and exited the case.  Those he did resolder and insulate.  I don't think that he has used a meter on anything, though

What I was wondering was....could this be a symptom of a missing ground connection somewhere?  Static and/or high tension voltage can act strangely sometimes.

Greetings gentlemen (and ladies). I now am an official member. Your collective contribution to my ignition problem is greatly appreciated. The dialogue to date depicting the ignition problem is accurate. I have again re-checked all connections and cleaned and grounded the various ignition components. When I opened the left side case again, I did measure the pick-up's resistance and it is within tolerance (200 ohms or so). I soldered and sealed the minute insulation cracks on the G and O wires at the grommet. I saw no visible damage to the pick-up in terms of debris from the broken gears. And I checked for continuity on all ignition related wiring between components (switches, ignitor, coil, and regulator). After all this; still only one strong blue spark on the grounded spark plug on the first revolution of each cranking attempt. I will go back, as you suggested, and check for voltage in and out of the black box.
Greg: Thanks for getting this open dialogue initiated. I don't know of any other Suzuki owners in the area (Montreal, Quebec) in an attempt to swap parts for testing.
I sincerely welcome any more thoughts on the topic. Thanks.
Derek

The problem persists.
I have gone through Slavy's step by step suggestions and all connections, resistances, current, voltages, and grounds appear as they should be. So I decided to investigate a bit further. When disconnecting the coil there was about 5ohms across the terminals. When connected, the orange wire showed 12v with the key on. But when cranking the engine, there was 10.5-11.5v at the terminal. As there is no balast resistor in the system, I assume that the minor voltage drop simply is from the starter drawing power. If my memory of electrical theory is correct, the coil should receive 12v to excite the field and the magnetic field should collapse in between excitations/modulations. But, the needle on the voltmeter did not fluctuate when cranking. Now I'm suspecting the signal is corrupt from the black box which I understand controls this.
There continues to be one spark at the beginning of each starter cycle attempt. A bizarre observation though is that when I turned off the key, there is another single spark. Is the black box all confused? I don't think this is normal. I rechecked all possible sources of power interruptions and the kill, neutral, clutch, and side stand switches are operational. So, I'm thinking that the ignitor box may be bad. The bike did however sit outside over the winter at -30C and may have cracked internally the black box. Poor Savage. At 650$ I'm not rushing out to buy a new black box right now. If you have thoughts and/or have a comment on my deductions and attempts at problem solving, please do share. If there is something else to check, again, please do share. Thanks.
Derek

99% sure it is the CDI. Before You buy a CDI just verify that the pick up coil is good.
There are CDI'S on eBay for about $40. Just be patient. The 86-88  CDI is different than 95-2004. It might be just the connector , but I am not sure.

You can check everything else, exept the CDI. If everything else checks good- blame it on the computer.

I got one more thing to clean up here.... just in case....

The thing was sitting outside in the weather? I beleive I read that?

The flywheel could have devloped rust and corrosion. Interfering with the CDI to HES interaction. I would clean the flywheel inner side and the HES contact as well as the contact surfaces of the Stator.

I am just grasping at straws here but weird things happen with a coat of rusty corrosive slop between these two important points that feed the CDI and other things. This is the point of generation for all that power needed to supply those kinds of things.

You should be able to do this without pulling the stator off but just pulling the flywheel. Make sure it gets proper keyed and tightened though.