"The first use of harebrained dates to 1548. The spelling hairbrained also has a long history, going back to the 1500s when hair was a variant spelling of hare. The hair variant was preserved in Scotland into the 18th century, and as a result it is impossible to tell exactly when people began writing hairbrained in the belief that the word means "having a hair-sized brain" rather than "with no more sense than a hare." While hairbrained continues to be used and confused, it should be avoided in favor of harebrained which has been established as the correct spelling."
FYI, here is the usage note from AHD:
"The first use of harebrained dates to 1548. The spelling hairbrained also has a long history, going back to the 1500s when hair was a variant spelling of hare. The hair variant was preserved in Scotland into the 18th century, and as a result it is impossible to tell exactly when people began writing hairbrained in the belief that the word means "having a hair-sized brain" rather than "with no more sense than a hare." While hairbrained continues to be used and confused, it should be avoided in favor of harebrained which has been established as the correct spelling."
not hairbrained