The capacity of the battery has a lot to do with its size; you can't get away from that. Some technologies are better than others; various lithium rechargeables are some of the best, but charging circuits aren't as easy as NiMH.
But, to the project: First define how bright it needs to be (including the diffuser). Then, choose an LED. Try to find the most efficient that you can. Define operating time. From all this you can calculate how much energy the battery needs to hold. (volts x amps, x hours). From there, choose a battery, and then a driving circuit (if you have a 3V LED but a 1.25V battery seems to be the best fit, there are booster circuits.)