Tried a bunch of stuff with the Arduino, including Shonky's suggestion, but got no improvement.

I tried putting the Arduino actually to "sleep" with its own sleep commands and got no improvement. Which surprised me. I would have thought that if I could make the noise go away and come back by moving the wires farther and closer to the Arduino, that it had to be the Arduino. But now I'm wondering if the interference is coming from some other part of the Arduino's board. Such as its power supply or its clock crystal or something like that.

Shonky, if I had a 'scope, what would I check? Maybe I'll invest in one of those iphone 'scope attachment doohickeys.