In 2005 Suzuki switched to a system that incorporated two sets of contacts in the starter button switch.  One set of contacts closes when the button is pushed, and the other set of contacts opens when the button is pushed.  The set of contacts that open when the button is pushed provide power to the lights and horn.  This feature is intended to provide more juice to the starter during the start cycle.  It shuts off the lights when you are trying to start the engine.

The power to your lights and horn runs through the starter button switch.  When the button is in the released position, power from the orange wire is connected to the yellow/white wire.  The yellow/white wire runs from the starter button to the multi-connector under the tank, where the yellow/white wire is connected to black/red wire.  The black/red wire hots up the horn and the lights.

Check your starter button switch.  Verify that you are getting 12V at the yellow/white wire when the button is in the released position.  If you are getting power at the yellow/white wire, check the large multii-connector under the tank and verify that you are getting 12V to the black/red wire.

Had to get a new meter - but everything looks ok. Now what?  

Yes, description unclear.  Odysseus, are you saying the motorcycle starts and runs, but the lights and horn don't work?  That is the impression I get from your description.

I just did a quick & dirty check on mine.  Disconnected the red/black wire from the horn and connected to meter (+).  Connected meter (-) to engine ground.  Key on meter reads +10.5V.  Press start button meter reads 0.0V.  You should do same test.  If you don't get voltage to horn connector with key on and start button released, use the attached diagram to ring the system out.  The circuit runs through both multi-connectors under the tank (right hand and left hand).

Ok, I am starting to track this down. I've traced the wire back to the black and red wire by disconnecting the horn/light mechanism from its position in the box as shown in the pics. There is a general view of the box and a close up of the red / black wire's beginning. The third pic is under the seat. So, from here do I simply track this wire? Because to do that would require cutting open a bunch of tape and black wire coverings.

Here's the under the seat view....

Since it DID work once or twice but failed again, I am now starting to wonder if I have a 20-yr old funky connection that requires cleaning somewhere maybe. Maybe.

Most connectors have a ground wire, black w/a white stripe as well as frame ground.

The main ground from the battery is attached to the right rear of the engine just behind the oil level sight window.

If your engine runs but the horn and lights don't work, then you can check to see if you are getting voltage past the starter button by doing a quick and dirty test.  I already mentioned this in one of my prior posts.  This video shows how to do the test.  I think you will see how easy it is.  The switch incorporated into your starter button is pretty wimpy and it gets fouled with rain and dirt.  I have had two of them fail; one on my LS, and the other on my AN.  They both start getting intermittent.  The lights and horn fail to come back on when I release the starter button.  If that's how yours is behaving, if the engine starts and runs but sometimes the lights and horn don't work, do this test.  You can repeatedly press the starter button to verify that the contacts in the switch are good.

https://www.youtube.com/watch?v=mKZnMXi0h60