Thanks Gerald and Dave for your suggestions.
Gerald: Do you mean a ground connection in addition to that provided by the serial port?
I'm using an FTDI breakout board from SparkFun, and I have a ground wire running from the bbmx pin 5 to the ground pin on the breakout board. (The only other wires are pins 2 and 3.) This is connected to my laptop via a USB cable, presumably with ground.
I have no power ground because the bbmx is powered by a BeagleJuice battery. All of my power supplies are double insulated, as Dave suggested, as is my laptop supply.
What I did do is connect another usb cable to my computer, and then hold the shell of the other end against the metal shell of the connectors on the bbmx. Assuming all metal shells are grounded, this should have solved that problem.
No difference.
I also tried another serial adapter; this one a Prolific cable. Same thing.
I'm busy rebuilding my sd card right now (downloading Angstrom-Beagleboard-demo-image-glibc-ipk-2011.1-beagleboard.rootfs.tar.bz2 is time consuming!) and will report on that.
Meanwhile, further information:
On reset, the two green LEDs next to the uSD card (USR0 and USR1) go out. THey both come on after about a second.
After about 10 seconds they both go out briefly, then they have a burst of activity, and then USR1 stays on while USR0 does a beat of either 2 or 3 flashes.
Every so often USR1 gets a burst of activity, and this seems to correspond to a burst of garbage on minicom.
Does this suggest anything?