What earthquake allowed researchers to develop the elastic rebound theory?

The answer is: the 1906 San Francisco earthquake.

From an examination of the displacement of the ground surface which accompanied the 1906 earthquake, Henry Fielding Reid, Professor of Geology at Johns Hopkins University, concluded that the earthquake must have involved an "elastic rebound" of previously stored elastic stress.

The elastic rebound theory is an explanation for how energy is spread during earthquakes. As rocks on opposite sides of a fault are subjected to force and shift, they accumulate energy and slowly deform until their internal strength is exceeded.