When someone refers to someone else as “harebrained,” they mean that the harebrained individual has little more sense than a hare, particularly a hare during mating season. Hares, close relatives of rabbits, are legendary for being extremely flighty and not terribly bright. A similar slang term is “birdbrained,” which means roughly the same thing, and both are generally viewed as uncomplimentary.
The first documented use of “harebrained” dates back to 1548, according to the Oxford English Dictionary. However, the variant “hairbrained” dates from around the same time. In fact, the variant spelling is not a totally different slang term meant to suggest that someone's brain is made from hair. It's actually a reflection of the dispute over spelling “hare,” which endured well through the 1800s in some English-speaking nations. Most people view “hairbrained” as incorrect.
A harebrained person is foolish, flighty, thoughtless, stupid, or reckless. The general idea is that the person is clearly lacking in sense, and perhaps even preposterously stupid. Some other synonyms for “harebrained” include: dopey, dotty, loony, crazy, foolish, softheaded, imbecilic, mad, and sappy, among others. The term is also used to refer to someone with a vacant, airheaded, frivolous personality, reflecting a lack of gravity and seriousness, as well as a disconnection with reality.
In addition to people, plans and schemes may also be referred to as “harebrained.” In these instances, while the originator of the plan may have some redeeming qualities, his or her plan leaves much to be desired. Some people may also suggest that behavior which was totally unplanned was the result of a harebrained scheme, in a way which is meant to gently remind people of the importance of thinking ahead. People also sometimes use the term “harebrained scheme” to refer to their own plans in a self-deprecating way, to suggest that they know a scheme sounds foolish, but they think it might actually work.
In fact, hares are probably no more or no less bright than other wild animals. The behavior of hares during mating season might seem erratic or even crazed to human observers, but hares undoubtedly have their own agendas, and whatever they are doing seems to be working, as they are in no danger of extinction. The flighty nature of hares is undoubtedly a protective response, as a high sensitivity to predators and the ability to run and hide quickly would ensure that an animal is more likely to survive attacks.