Is that proposal to raise your taxes harebrained or hairbrained? Harebrained means "having or showing no more sense than a hare." It is a well-established word. Its first use dates to 1548. The spelling hairbrained also goes back to the 1500s, when hair was a variant spelling of hare. The hair variant was preserved in Scotland into the 18th century. The upshot of this is that it's impossible to tell exactly when people began writing hairbrained in the belief that the word means "having a hair-sized brain," but it's a good guess that it was a long time ago. Whatever the date of the first confused use of hairbrained, it continues to be used and confused to this day. If you want to give the impression that your brain is bigger than a hair and bigger than a hare's, use harebrained.
COCKAMAMIE. "'Cockamamie' means something worthless or trifling, even absurd or strange; a 'cockamamie' excuse or story is an implausible, ridiculous one. The word may be a corruption of 'decalcomania' ('a cheap picture or design on specially prepared paper that is transferred to china, wood, etc.'), a word youngsters on New York's Lower East Side early in the century found tiring to pronounce and impossible to spell." From "Encyclopedia of Word and Phrase Origins" by Robert Hendrickson (Fact on File, New York, 1997).
"Cockamamie is the child's version of 'decalcomania,' dye transfers that youngsters used to put on their hands and arms. Since they were cheap, they soon wore off. So, 'cockamamie' first meant anything trifling or second-rate, and later came to mean simply silly or laughable." From "Morris Dictionary of Word and Phrase Origins" by William and Mary Morris (HarperCollins, New York, 1977, 1988).
"Cockamamie or cockamamy - adj. Slang. 1960, from earlier cockamamie decal (probably before 1926), apparently an alteration of decalcomania." From "The Barnhart Concise Dictionary of Etymology" by Robert K. Barnhart (HarperCollins