In which direction would the ball bounce off to? Given is the following illustration of an arm rotating clockwise centred around the black dot. The top bit of the arm will hit the red ball. In which direction would the ball shoot off to?
My intuition tells me it would be in the direction of the normal of the arm, is this correct?

 A: The ball will only move in the direction of the normal if it is approached along the normal. Otherwise it will bounce off at an angle, following the Law of Reflection : angle of incidence = angle of reflection. This assumes there is no friction between the ball and bat. 
Such slice shots occur in tennis and golf : the direction of motion of the bat is not along the normal to the face of the bat. In your diagram the direction of the normal is changing as the bat rotates, but it looks like the collision happens along the normal, so the ball will move in the direction of the normal at the point of the collision.
This is a collision so the conservation of momentum applies, also the Law of Restitution which says that the relative speed of separation equals some constant $e$ times the relative speed of approach. $e$ depends on the nature of the materials and tells you what fraction of kinetic energy is lost in the collision.
You can look at the collision between the bat and the ball from the perspective of the bat. Then the ball approaches the stationary bat in the opposite direction but with the same relative velocity. Assuming the bat is much heavier than the ball, the ball bounces off the bat as it would off a wall. The component of velocity of the ball normal to the bat is reversed, the component parallel to the bat is unchanged (assuming no friction).
If the collision happens along the normal and no kinetic energy is lost (ie this is an 'elastic' collision), and the bat is much heaver than the ball, then the ball rebounds with twice the speed of the bat.
If there is friction the ball is reflected closer to the normal : angle of incidence > angle of reflection. If the ball is rotating the collision is more complicated.
