Which property of a wave relates to its height from the rest position?

The property of a wave that relates to its height from the rest position is amplitude. Amplitude is defined as the maximum distance that points on a wave, such as the crest or trough, move away from their rest position or equilibrium. This gives a measure of the energy carried by the wave; higher amplitudes indicate more energy.

In contrast, wavelength refers to the distance between successive crests or troughs in a wave, frequency concerns how many cycles or waves occur in a given time period, and phase pertains to the position of a point in time on a waveform cycle. The focus on height in this question correctly identifies amplitude as the property that corresponds specifically to this measurement.