The rubber band only has potential energy if it is stretched, so you work out the potential energy by observing the way it is stretched and how much force it took to do it.
The following applies to springs, but it should be about right for an elastic band.
In the case of a spring of natural length l and modulus of elasticity λ under an extension of x, elastic potential energy can be calculated using the formula:
Potential energy E = (Modulus of elasticity) times (extension)^2 divided by (2 * natural length)
Modulus of Elasticity is Stress / Strain.
You would have to calibrate your elastic band by stretching it with a known force (hang a 1 Kg weight off it and see how much longer it gets) to get a figure for Modulus of Elasticity.
.To confirm the potential energy, you could use the extended elastic band to fire a projectile vertically upwards.
Something like a pencil or a satay stick could be weighed and then fired.
If you could estimate the height (by recruiting helpers to observe how high it goes) you could calculate the initial velocity and convert this into Kinetic Energy by using
KE=1/2 M V^2. This is approximately equal to the potential energy of the elastic band before it was used to fire the projectile.