What is the name of the part of the temporal bone that houses the organs of hearing and equilibrium?

The petrous part.

Explanation:

The petrous part of the temporal bone has the shape of a quadrangular pyramind whose axis, from base to vertex, is oriented obliquely forward and inward. This triangular portion is located between the occipital bone and the sphenoid. It houses the organs involved in hearing and balance/equilibrium (This part of the bone contains the inner ear). It has three surfaces, three angles, a base and a vertex, formed by the inner part of the petrosal portion and the fetal tympanic portion. This portion contains numerous cavities and multiple ducts that cross it. The base of the petrosal portion is intended to form the mastoid portion, serves as the medial wall of the middle ear cavity and provides an outlet for the Eustachian duct.