The expression “as tight as Dick’s hatband” means: Absurdly queer, or, as the case may be, inordinately tight.
The “Dick” alluded to in this metaphor was Richard Cromwell, “Lord Protector” of England for a few months, September 1658 to May 1659.
He had been nominated by his father, the powerful Oliver Cromwell, to succeed him in this high office, and was actually so proclaimed.
But whereas the father had served, at least from the death of Charles I in 1649, as quasi-king of England, king in fact if not in name, Richard would gladly have accepted both title and crown, had not the army been hostile to such action and, indeed, to Richard, who was shortly dismissed from office.
The crown was the “hatband” in the saying, which was deemed a “queer” adornment for the head of one so briefly in highest office, and too “tight” for him to have worn in safety.
Let us add, however, this account is not accepted by the Oxford English Dictionary, though no better substitute is offered.