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Mass
Mass can be an illusive idea and difficult to separate from related concepts such as weight, volume and density. Where the pendulum is concerned, the mass of the pendulum --- or the mass at the end of the pendulum --- can be thought of solely as its weight. A more technical and accurate description would also consider its density.
The Pendulum
You can think of weight as the degree to which an object is pulled towards the center of the earth by gravity. When the weight on a pendulum is rotated along the radius of the pendulum and released, its weight is pulled by gravity, but the force of gravity is overcome by the mechanical force of the pendulum. Instead of traveling straight towards the center of the earth, the gravity and the mechanic restraint of the pendulum both affect the weight, moving it along an arc.
The Law of Conservation of Energy
Excluding considerations of wind resistance and friction, heavy objects drop at the same speed as light objects. So, there is little visible difference between a heavy pendulum and a light one, but there are differences. Imagine the full swing of the pendulum; then divide it in halves with a vertical line. Heavier pendulums will have more rotational inertia as they cross this imaginary line. In this short arc, it will gain inertia slightly faster by overcoming friction and wind resistance with greater ease. It will also decelerate slightly faster because gravity will act upon its upswing in proportion to its weight. Overall, it will cycle a little faster, not because of the weight but because of the relationship between its inertia and the forces that impede its progress: friction and air resistance.
Rotational Inertia
At first, the arc moves the weight approximately towards the center of the earth. Once the armature of the pendulum reaches perpendicular to level, the weight begins to climb. Inertia, in this case rotational inertia, overcomes the force of gravity to cause the pendulum's ascent. Gravity and rotational inertia come to equilibrium, before the pendulum falls again in the opposite direction. High rotational inertia isn't visible. You can think of it as a kind of latent or stored energy. If there were a collision, you could see it. For example, if you placed a tin can in the center of a 6 oz. pendulum and compared it to an identical tin can in the path of a 24 oz. pendulum, you could easily see the greater rational inertia transfers into the can in a collision.