I wanted to test the bulb before I installed it.  No point in fitting it to the 'scope if the AA batteries won't power it.  I don't have a power-source.  So I scrounged some speaker-wire, taped the AA's together, and taped the speaker-wire onto the ends of the "battery pack."  The voltmeter confirmed I had 3V power at the end of my test leads.  Just for giggles, I applied the leads to the old blue bulb.  As I very much expected, the old bulb didn't light.  (Which is good -- now Nicole won't kill me for purchasing an unnecessary replacement LED.)

When I touched the leads to the LED, it lit.  Huzzah!  I have a working bulb, that can be powered by the 2 AA batteries that fit into the microscope's base.

So why light?!?  That's the LED, installed in the microscope.  The batteries are in place.  I confirmed I have power at the bulb contacts, and at the bulb.  It's the same batteries, the same LED -- the only difference is the wires.  Instead of the test leads, power's through the wires in the microscope.  And those are working, because I have power at the contacts.

I just don't understand it.