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"I am as queer as Dick's hatband; that is, out of spirits, or don't know what ails me."
--A Classical Dictionary of the Vulgar Tongue, by Francis Grose (1796).

'Dick's hatband' was a very popular expression, apparently originating in England, but spreading quick to America. It is, unfortunately, also a quite mysterious expression. The first written documentation comes from a 1794 English-German dictionary:

"Queer as Dick's Hatband: in Unordnung, ohne zu wissen, was einem fehlet." (trans. 'in disarray without knowing what is wrong with you.')
--Vollständiges Wörterbuch der Englischen Sprache für die Deutschen, Vol. 2, by Johann Ebers (1794).

We must assume that by the late 1700s it was a fairly popular saying, but two mysteries remain: who was Dick, and what was up with his hatband? People love making up etymologies, so we have the comparatively popular and completely unsubstantiated theory that the Dick in question is Richard Cromwell, who was briefly Lord Protector of England in 1658, and the hatband was his crown. He was not well fit for the post, and lasted only eight months.

Other theories include that Dick is a variant on Nick or Dickens, that is, the devil, or that Dick was simply "some local character or half-wit" (OED). This is not a bad guess, as Richard was an extremely common name, and even when the phrase was in common usage nobody really knew who Dick was. This, as it turns out, was a benefit, because it meant that Dick's hatband could be whatever you needed it to be:

"DICK’S HATBAND: This very singular expression I have often heard in Rhode Island. {...} It is in general use throughout Shropshire, where it is applied as a comparison for what is obstinate and perverse. Ex. 'As curst as Dick’s hatband, which will come nineteen times round and won’t tie at last'; 'As contrary as Dick’s hatband'; 'As false as Dick’s hatband'; 'As cruikit as Dick’s hatband'; 'As twisted as Dick’s hatband'; 'All across, like Dick’s hatband'; 'As queer as Dick’s hatband'.
--A Dictionary of Americanisms, by John Russell Bartlett (1848).

Perhaps most common today, at least in America, is "as tight as Dick's hatband", but for the last hundred and fifty years, at the least, 'Dick's hatband' has simply been an intensifier, and can be used with any adjective -- as evidenced by the aberrant forms "as plain as Dick’s hatband" (2019), and "as fine as Dick’s hatband" (1898).