Does rotational inertia depend on the mass of an object?

yes, the rotational inertia does depend on the mass of the object.

An object gains rotational inertia by having mass at a distance from a particular point of rotation. It is not simply the mass of the object, but how the mass is distributed and the point of rotation.

for example, if you put a mass on the end of a meter stick, it is easy to make the meter stick rotate if you hold it at the end where the mass is (low rotational inertia), but it is difficult to get it to rotate if you hold it at the end opposite the mass. The greater the mass, the greater the inertia, but placement counts too.