It looks like I have, at long last, solved the problem on my 2000 Sienna. Apparently, the intermittent horn was due to a grounding problem, as others here have described. An automotive electrician ended up running a wire from the ground wire of the spiral cable that goes to the cruise control system directly to the steering column shaft and that fixed it. He said the shaft ground was interimittent and suggested something about the tilt mechanism, although I may have misunderstood the tilt comment. He said it was a bad design to have the grounding set up the way it was, just as Dave has suggested.
Unfortunately, before doing that, he convinced himself (and still remains convinced) that the spiral cable was bad and first replaced that. When the interimittency returned, he assured me that the sprial cable was bad in addition to something else, the something else being the grounding problem he later found. I think he is convinced that the spiral cable was bad (I could not fully understand the tests he ran to convince himself of that), not just becasue, if it wasn't, he would have had to eat the expensive cost of the part. Between the visit to diagnose, then the second visit after the special order spiral cable came in, plus the 3rd visit to fix the ground, they must have spent 8 hrs labor, for which they could only charge me their original $140. Aftre he fixed the ground, he just did not want to try it with the old spiral cable, since the problem had been interimittent, and I was just happy that they were finally done, having watched them for 6 of the 8 hours. So, all in all, I'm not unhappy with the result.
Anyway, I thought I should add my voice to the chorus of grounding problems in these Sienna models, and perhaps other Toyotas. It seems that the problems started after the vehicle had been driven 90k miles and was initially temperature and vibration related, later only vibration related. Ultimately, with the steering pad disassembled, he could reproduce it by minute movements of the steering wheel.