Many expressions used the word "mad". One is "mad as a hatter." Another is "mad as a March hare." Both these expressions are based on the same children's book - Lewis Carroll's Alice's Adventures in Wonderland. The story is about Alice, a girl who falls through a hole in the ground into a very unusual place. Alice meets strange people and animals, including a disappearing cat. The cat tells Alice, "In that direction lives a hatter. And in that direction lives a March hare. Visit either you would like. They both are mad."
Alice has tea with the Hatter and the March hare. It is the strangest tea party you could imagine. Alice decides that the cat was right. Both the Hatter and the March hare are mad. Alice in Wonderland is a make-believe story. But some of it was based on fact. Hatters - people who made hats - used the element mercury in making and shaping hats. After years od working with mercury, many hat-makers suffered from mercury poisoning. They lost control of their muscles and many became insane. So, there was a reason for the expression "mad as a hatter."
"Mad as March hare" is even easier to explain. In the spring - usually in the month of March - hares, or rabbits appear to go insane. They can be seen in fields hitting the ground with their back legs. Jumping up and down. Throwing their bodies in circles in the air. The hares appears to be insane. But they are only doing what comes naturally. So, the hare is not mad all the time... just in the month of March.
Another expression, "mad as a wet hen" is not about insanity. It is about anger. People use the expression to describe someone who is really angry.