harebrained » hairbrained
The involvement of rabbits in this idiom seems to be non-obvious to a great many speakers; “hairbrained” gets ca. 10k raw Google web hits to ca. 50k for “harebrained” (1 April 2005). The much more frequent “hair” is giving the rarer “hare” a serious race for its money. The semantics seems to involve hair as inconsequential, fluffy, silly.
See also other “hare”/”hair”-related entries.
James Cochrane, Between You and I, labels this a spelling error. Perhaps it is for some people.