I'm not entirely satified with this answer, but here's what I found at http://www.amerispeak.com/June%2098.htm
Origin: hare + brained. A hare is a rabbit-like animal with long, floppy ears. It is known for its jumping ability, not for its intelligence. Thus, harebrained = foolish, silly, not very smart or well-thought out.
Harebrained is an adjective. It is often used to describe an idea.
a harebrained idea
a harebrained notion
a harebrained scheme
The main reason I'm not satisfied with this answer is my agreement with VilleOrb where he said "Hare's aren't the smartest creatures but why pick on them?"
"I wept because I had no shoes, then I met a man with no feet. So I took his shoes" - Dave Barry