Ok. Two nice bright LEDs, plus a little shielding around the light sensor, did the trick. I now get a difference between nothing and something in the mailbox that exceeds 3 times the sensor threshold. I.e., with the threshold set at 3, I see SensorValues with nothing present of 10. But, with a small postcard present, the SensorValue jumps to 18 or greater. So, no ambiguity and no thrashing web service output.
Now, a new thought. Instead of providing the +5vcc to the 1127 from the Analog input of the ifKit, I could control it from a Digital Output. This way, instead of leaving the 1127 and LEDs on constantly, I could turn the 1127 on at regular intervals, say once a minute or 5 minutes, take a reading, and then turn it off. I assume I'd have to use a relay to control the 5v power from the ifKit. Does this make sense?