The project got brought back from the dead. Turned out what I had thought to be poor tuner performance was something awry with the offending station.

Got the Flirc app running on my Linux box this morning.  Plugged in the Flirc, fired-up the app.  The app saw the Flirc and notified me of an available firmware update.  Installed it.

Programmed the MX-500 from a canned NEC TV profile (first one I tried that produced signals for all the joystick buttons: TV 130).  Taught the Flirc from it.  Plugged the Flirc into the MiBox.  Worked!

Taught the MX-500 the MiBox' power on/off from the inexpensive GE/Jasco remote.  Done!

Only MiBox button I'm missing I would like to have is the "Home" button.  Wouldn't hurt to have the MiBox' volume up/down buttons, as well.

Btw, m1964: You can teach the MX-500 the Roku box from that inexpensive GE/Jasco remote, including the elusive "*" button that, everywhere else on the 'net, they say can't be taught/learned to/on an IR remote :)

ETA: Got the "Home" button by switching to the Fire TV template and teaching the MX-500's "Menu" button to "Home" on the Flirc. Life is good :)