E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
A Tom Noddy is a very foolish or half-witted person, a noodle. The marine birds called Noddies are so silly that anyone can go up to them and knock them down with a stick. A donkey is called a Neddy Noddy.
Minshew has a capital guess derivation, well fitted for a Dictionary of Fable. He says, Noddy, a fool, so called because he nods his head when he ought to speak. Just as well derive wise-man from why, because he wants to know the why of everything.