1A person who scratches his or her head, especially in puzzlement or thought; (hence) a perplexed or thoughtful person. Compare to scratch one's head.
2US informal A perplexing problem or question; something which causes bafflement or puzzlement.
3An implement for scratching the head. Now chiefly historical.
Early 17th century; earliest use found in Barten Holyday (1593–1661), Church of England clergyman and poet. From head + scratcher.