I was going for a waterfall / flowing effect...but the result didn't quite look like it.
The CD4040BE ripple counter can be used as a frequency divider and the badge uses the first four divisions. The 555 timer trigger and threshold pins are tied together. A CdS photocell (of approx 25K-500K ohms) is in series in the "discharge half" of the RC timing circuit. The 39K ohm resistor serves to create a floor on the timing. The capacitor in the timing circuit is a 1uF electrolytic. There's a mistake in my quick schematic (in the photos) that I luckily did not transfer to the badge (the reset pin should be tied to +3V).
The neat thing about the 4040 is that it has buffered outputs that can drive or sink current with a current limit (based on the voltage). So, if you drive it from 3V coin cell, apply the eyeball-fudge method on the datasheet graph, the outputs are limited to about 12-13 mA. This is perfect for the 3mm through-hole LEDs (but not enough for old 5mm ones that usually need 20-25mA, they were pretty dim). Additionally, because you can both drive and sink current on the outputs, you can tie the cathode of one LED and the anode of another to the same output to get alternating blinkenlights.
The 555 timer is one from a small stock of them I've had since my uncle gave them to me in the 1980s.