Say you have two objects colliding, and there is some elasticity between them. Some of the kinetic energy of the objects will change into elastic potential energy when they collide, but when they bounce off each other again, the energy does not return to being kinetic. What happens with it? Does it stay stored as elastic potential energy, or does it change to thermal energy? Or does it perhaps do something entirely else?
Sounds like you're getting at the "coefficient of elasticity," which is a value in [0,1] which represents what percent of the pre-collision kinetic energy is found after the collision. In homogeneous materials, the remainder of the energy is typically lost to deformation or heat (phonons) as you suggest.
you could imagine, for the sake of argument, a steel ball hitting an object which has a spring with a retention device (a locking lever of some sort). In this specialized case, the steel ball does in fact transfer a decent amount of recoverable potential energy into compressing the spring. It takes some other action, i.e. releasing the spring, to return that portion of the potential energy to kinetic. To be clear: suppose the steel ball has N joules of kinetic energy but the spring bottoms out at M
but when they bounce off each other again, the energy does not return to being kinetic
The potential energy actually does go back to kinetic energy. But often it only partially goes to the translational motion of whole body, while other part of the energy comes into internal motion of atoms. People normally call the latter thermal energy.