Throw your hat into the ring

The ring in question here is a boxing ring. These, of course used to be circular spaces in a crowd of onlookers, rather than the square, roped ‘rings’ of contemporary pugilism. Any Jack the lad who fancied his chances in a bout would throw in his hat – presumably this was a more

reliable way of putting oneself forward than just shouting over the hubbub of the crowd.

The expression dates from at least the early 19th century. The earliest citation of it that I have found is from an 1805 issue of The Sporting Magazine, or as the publishers preferred to call it The Sporting Magazine or Monthly Calendar of the Transactions of the Turf, the Chace, and every other Diversion Interesting to the Man of Pleasure, Enterprise and Spirit:

Belcher appeared confident of success [in a boxing match], and threw his hat into the ring, as an act of defiance to his antagonist

That citation doesn’t specifically refer to a challenge. Another reference, from just a few years later, supplies that – The Mirror of Taste, published in Philadelphia in 1810:

A young fellow threw his hat into the ring and followed, when the lame umpire called out “a challenge,” and proceeded to equip the challenger for the game. … He then walked round the ring till a second hat was thrown in, and the umpire called out, “the challenge is answered.”

See also: throw in the towel.