I salvaged a solar garden light from the Vassar Community Gardens midden heap and stripped it down:

The single IC is a YX805 “solar lawn light boost control chip” and the resistor-like thing is a 82 µH inductor setting a 13 mA input current.
Cleaning off some minor corrosion, charging the NiMH cell, and soldering an amber LED onto the pigtail wire brought it back to life.
It’s now perched on the porch railing where it catches some afternoon rays:

Maybe we can think of something better for it to do …