Check out MANWARING, MANNERING, etc.
Don't forget that spellings change over time. The original surname was closer to MANWARING (some people say it came literally from someone who was a crew memebr on a "man-o'-war" sailing ship).
The 'w' got lost, and many started spelling their name MANORING or MANNERING. Since the 'e' in the middle had very little emphasis when speaking the name, it could have been lost, to arrive at MANNRING or MANRING.
Look up these other spellings on this and other forums.