Newbie question: Light weight, temporary power?
Adapting How long would a 150 Farad Capacitor light up an LED (answer: about an hour), we would get that a 2.5F capacitor would light an LED for a minute. They exist, but are still pretty bulky.

I'd just stick with an appropriately tiny coin cell. If you want to overdo it you could have a Li-ion which will last for hours, but they're a bit fragile to put in darts.