As mad as a March hare
Hares have long been thought to behave excitedly in March, which is their mating season. Lewis Carroll is among many who have used this idea in stories - Alice's Adventures in Wonderland:
"The March Hare ... as this is May, it won't be raving mad - at least not so mad as it was in March."
More recently this behaviour has been questioned and it is now thought that hares behave oddly - boxing, jumping etc. - throughout their breeding season, which extends over several months.
Be that as it may, hares, especially March hares, have that reputation, which will surely stay with them.
The first record of the belief in their madness, or in this case their brainlessness, was circa 1500, in Blowbol's Test reprinted by W. C. Hazlitt in Remains Early Popular Poetry of England, 1864:
"Thanne [th]ey begyn to swere and to stare, And be as braynles as a Marshe hare."
Of course, the phrase 'hare brained' refers to the same behaviour. This is also old and is referenced in Edward Hall's Chronicle, 1548:
"My desire is that none of you be so unadvised or harebrained as to be the occasion that ..."
The first citation that uses the phrase in a form we now know it is in 1529, in Sir Thomas More's The supplycacyon of soulys:
"As mad not as a march hare, but as a madde dogge."
The phrase has been in continuous use in the language since the 16th century. It was well-enough established by 1546 for John Heywood to include it in his collection - A dialogue conteinyng the nomber in effect of all the prouerbes in the Englishe tongue.
See other 'as x as y similes'.
See 'as mad as a hatter'.