I didn't expect this to attract as much attention as it did.  I have a lot of reading to do on negative resistance devices.  I caught a cold in the last couple days, so I haven't had much time or energy to do any further experiments, but I do want to make measurements across the different junctions and try to characterize what is happening.

I know it's not a very practical circuit, and probably won't be reliable, but I like strange, goofy things.  I found that there were multiple attempts by different companies to create 1-Transistor SRAM... everything from self refreshing DRAM that pretends to be SRAM, to SCR and Thyristor based designs, and they all kinda didn't work out as expected.

I have found this so far:

1. The off state isn't really off.  If I turn the lights out the LED glows very dimly.  The circuit just has a high power and low power state.
2. The circuit works without the LED.  If remove the LED, and drop the supply a couple volts, a voltmeter will read two voltages for the two different latched states.  So there's no weird interaction with the diode; it is really truly a single active transistor.
3. I left it on the off state overnight and it was still off.  I turned it on, and left it another day and it was still on.  I didn't watch it in between, but it seems very stable.
4. I have not ruled out thermal effects.  Overnight the temperature in the house changed several degrees.  But, I have found if the supply voltage is just right, and I have it in the 'off' latch state, if I remove the Rbc resistor, it does not turn on immediately.  It takes about half a second to move from off to on.  If I momentarily short the base to ground, it turns off, and then float the base, there is another half second delay.  There are no capacitors, except for whatever capacitance the transistor has, plus maybe the small amount the breadboard has.  But it doesn't seem enough to account for the delay.
5. If I put a capacitor in parallel with Rbc (say .1 uF), and measure its voltage in the on latched state, it wobbles quite a bit.  I've seen it go from .6 to .9 back to .6, over the course of a few seconds, but still keeping the LED on.  I have no idea what causes this random variation.  The power supply is a wall wart smoothed out with a LM317.  I would expect 120hz ripple if anything.
6. I have duplicated both the latching behavior and time delay with a battery supply.

I hope to revisit this sometime soon next week.  Maybe I'll try writing a paper for the hackaday journal.