Why does a solid rubber ball rebound from a hard floor?

Answer: The rubber deforms and returns stored elastic energy

Air beneath the ball is compressed and lifts it — Air squeezed beneath the ball is not the main source of its rebound. A solid rubber ball has no inflated inner chamber; most recoverable energy is stored when its rubber deforms during impact.

The hard floor stores most of the impact energy — A hard floor is not where most recoverable impact energy is stored in this case. It deforms much less than the rubber ball, so the ball's elastic recovery supplies most of the rebound energy. On a trampoline or another soft surface, the surface can contribute much more.

The rubber deforms and returns stored elastic energy ✓ — The rubber deforms on impact and stores elastic energy, then releases some of that energy as it springs back. The floor exerts an upward force while they are in contact. Energy lost to internal friction, heat and sound usually makes the next bounce lower.