The 16th century harebrained came from the earlier harebrain, "giddy or reckless person," from the common impression of a rabbit or a hare as being nervous and skittish.
What does it mean to be harebrained?
1 : foolish sense 1. 2 : absurd, ridiculous.
Is it hair brained or hare brained?
The spelling harebrained is most common. The spelling hair-brained is also frequently used, but some people consider it to be incorrect.