When a rubber band is stretched, it stores elastic potential energy, which theoretically increases its mass according to the principles of physics. This stored energy contributes to a slight increase in weight, similar to how a charged battery weighs more than a discharged one. The discussion encourages calculating the actual mass increase to understand the concept better. The energy stored in the stretched rubber band cannot dissipate, reinforcing the idea that it equates to mass. Overall, stretching a rubber band does result in a measurable increase in mass due to the energy stored within it.