Why is the acceleration of a pendulum non-zero at its lowest point?
There are two forces acting over the pendulum: the force due to gravity (weight), which always points downwards, and the tension of the rope, which is radial to the motion.
Tension is non-zero at the lowest point of the pendulum. Newton's second law states that a change in momentum (and by extension velocity) is caused by a force. The velocity of the oscillating body is changing as it passes through its lowest point: while the modulus remains constant, the direction of velocity is being changed upward. A force must be responsible for this change of velocity. The tension of the rope "pulls" the mass upward so that it follows with its motion.