I became intrigued, and decided to come up with some numbers. I hooked up a ROHM green 0603 LED part no. SML-310-MT and tried various currents in a darkened room. I only gave myself about 60 seconds to get "night vision".
Up close, I could 'find' the LED at 5µA if I already knew exactly where to look, but I could only focus on it at >20µA.
So, twelve of these babies plus three more for the hands are gonna need somewhere around 750 microwatts. So I guess 750µJ/s, or 2.7 Joules/hour.
So then let's see what a good capacitor can do. Sparkfun sells a 10F 2.5V capacitor for about $5; how much energy is that? I found an energy calculator which gives what seems to be a sensible result of 31.25 Joules.
If we can somehow fully charge two 10F capacitors in series to a total of 5V, they could theoretically run all 15 LEDs at 20µA each all night.
If we allow for half the charge (charge to 2.5V, discharge to 1.25V) and some losses, we might design with two or four of these caps. Each one is similar in size to an AA alkaline.
Incidentally, two AA alkalines would last about a year for a little over $1 but it wouldn't be as much fun.
Recharging requires finding a solar array that produces 5V at 2-5mA in room light. I think this could be the deal breaker.