Mad as a March Hare has been a common English phrase for hundreds of years. The phrase appears in John Heywood’s collection of proverbs published in 1546. It seems to have been inspired by the hares’ behavior at the beginning of the long breeding season, in March (the season lasts until September in Britain). Early in the season, unreceptive females often use their forelegs to repel overenthusiastic males. This energetic behaviour looks rather like boxing, and often includes vertical leaps into the air.
Blowbol's Test, a poem from about 1500, includes the line:
Thanne þey begyn to swere and to stare, And be as braynles as a Marshe hare
(Then they begin to swerve and to stare, And be as brainless as a March hare)