Small Footprint Single Pulse 500ms per minute LED 3.3v
An absolutely minimalistic solution is to drive a microcontroller from the button cell, and use a simple bootstrap circuit to drive the LED:







simulate this circuit &ndash; Schematic created using CircuitLab

You can program the MCU - represented by the dotted rectangle in the above diagram - to idle in power-save mode 55.5 seconds of each minute, and to toggle its output pin using PWM for the remaining 0.5 seconds. The PWM waveform on the base of C1 provides pulses of boosted current through the LED, D2, lighting it up.

If the LED's forward voltage is sufficiently low it glows dimly even when off, you may want to drive the anode of D1 from another IO pin so you can shut it off entirely when not in use.