An alternate explanation (which really is the same as the answer from @Phil): as per Kepler's laws, an orbit is an ellipse, and the orbiting period is proportional to the semi-major axis of the ellipse.
A satellite in the lowest orbit will try to follow a special kind of ellipse (namely, a circle), whose semi-major axis is really the Earth radius (this is the "lowest orbit" because the satellite grazes the ground -- we ignore the atmosphere here).
The oscillation in the hole is really another orbit -- it is a degenerate ellipse which has been flattened to a line. Yet its semi-major axis is still the Earth radius.
Same semi-major axis, hence same period.