I've been away for a while, but finally got back to this topic. Sergio's suggestion worked like a charm. Actually, I used a modification to the recipe to compress the image, since all I had was a 4GB flash drive (which was unfortunately just under the capacity of 4GB MMC on the Edison) and was too lazy to go out and buy a bigger drive. Specifically, what I did was:
On the Edison I wished to clone:
1. Create a place to mount a USB flash drive. I did 'mkdir /mount/usbdrive'
2. Plug in a USB drive to the USB OTG port, using a suitable OTG adapter cable and a powered USB hub.
3. Mount the USB drive: 'mount /dev/sda1 /media/usbdrive'
4. Create the compressed image: 'dd bs=4M if=/dev/mmcblk0 | gzip > /media/usbdrive/Edison_backup.gz'
Step 4 took about 35 minutes. (with bs=1M, it takes a lot longer)
On a brand-new Edison, right out of the box:
5. Connect via USB. I'm connecting with a Linux system, so: 'screen /dev/ttyUSB0 115200'
6. Create a place to mount a USB flash drive. I did 'mkdir /mount/usbdrive'
7. Plug in a USB drive to the USB OTG port, using a suitable OTG adapter cable and a powered USB hub.
8. Mount the USB drive: 'mount /dev/sda1 /media/usbdrive'
9. Transfer the image to the new Edison: 'gzip -dc /media/usbdrive/Edison_backup.gz | dd bs=4M of=/dev/mmcblk0'
Step 8 took another 35 minutes.
After step 9, the new Edison responds to every line command with a "segmentation error" indication. However, after a power cycle, the new Edison comes up as a perfect clone of the old Edison. It also got the partitioning right. On my old Edison, I flashed it with a 1GB root ('/') partition. The cloning resulted a 1GB root as well. The only difference was that when connecting to the new Edison wirelessly, my router assigned a different IP address to it, since it had a different MAC address.
In just a little over a half-hour, one can completely backup one's Edison, or in just a little over an hour completely clone it.
Thanks for the useful guidance, Sergio!
Jim