What happens when a photon hits a mirror?
How do mirrors work? is closely related to your question, if not a precise duplicate.
We normally think of photon scattering as absorbing the original photon and emitting a new one with a different momentum, so in your example of the mirror the incoming photon interacts with the free electrons in the metal and is absorbed. The oscillations of the free electrons then emit a new photon headed out from the mirror. Unlike e.g. electrons, photon number isn't conserved and photons can be created and destroyed whenever they interact.
If you think of this in terms of quantum field theory, which is really required to give meaning to the photon, then all you are able to say is that the photon can take any of all possible paths from where it is emitted to where it is absorbed. These paths will contain paths where the photon momentarily splits into an electron positron pair, where the interactions with the electrons in the mirror involve all sorts of virtual particles, where the photon travels in directions which are far from the classical trajectory etc. The total amplitude is given by the sum of all these possibilities and they can all occur. In the classical limit this sum over all paths gets dominated by the contributions closest to the classical straight line path of the photon with velocity $c$, so classically we see light travel in a straight line at velocity $c$, and obey the laws of optics. However if you really wanted to follow the path of an individual photon you would see that it could do any of a spectacular number of things (and unfortunately our attempts to observe the photon would interfere with its path). If you want to understand this better, I highly recommend Feynman's description of it all in his lectures here or in his book taken from the lectures: "QED, the strange theory of light and matter".
When considering the question "does the exact same photon of light bounce back or is it absorbed then one with the same properties emitted" it's important to remember that photons are indistinguishable bosons. The question suggests that such photons would be distinguishable, when they are not. In the end, It's just "a photon" with X momentum.
There is a meaningful distinction to be made about the reflection process though - There's no absorption and re-emission process going on here, as Anna V points out this would imply very different behavior. Photons and their evolution in time (up until, debatably, their measurement) is properly described as a wave, not as a particle which "bounces".
My classical picture is that when the incident electric field hits the metal (usually silver) surface of a mirror, the electrons, feeling the electric field, oscillate. As the electric field is oscillating very fast, the electrons tend oscillate out of phase with the field like a spring that's being wiggled too fast. The out of phase oscillation of the electrons produces a new wave which interferes deconstructively with the incident wave and cancels the field amplitude going into the metal, but produces a new field going away from the metal. Considering a large beam hitting a large area of a mirror, and therefore many electrons, all the electrons oscillate together and produce more collective interference effects, but the same picture holds. Tracking this down precisely to the quantum field theory level should be possible, although I personally would probably struggle to gain much intuition from those explanations.