The phrase to throw one’s hat in(to) the ring means to show willingness to enter into a contest or take up a challenge, especially in business or politics.

It was originally used in boxing with reference to the custom of throwing a hat into the ring to signal willingness to enter a contest; the earliest mention of this practice that I have found is from The Morning Chronicle (London) of Friday 30^th November 1804:

PUGILISM.

The fight which we stated a few days since to be about to take place between Tom Belcher, brother to the champion of that name, and Bill Ryan, son of the late noted pugilist, who fought with Johnson some years since, was yesterday decided at Wilsdon Green, on the Edgware Road, the spot where the hard battle was fought between Blake and Holmes, a twelvemonth since, and where Pictoun beat Will Wood in June last. A council was held among the gentry of the fist on Tuesday last, when the misunderstanding respecting the purse to be fought for was adjusted, and the champions agreed that the fight should take place yesterday, instead of Monday next. The champions arrived at Wilsdon Green at eleven o’clock in two hackney coaches. Belcher first threw his hat into the ring over the heads of the spectators, as an act of defiance to his antagonist, who received him in the ring with a welcome smile. The champions then stript and set to.
[&c.]

The earliest figurative use of to throw one’s hat in(to) the ring that I have found is from a letter published in the Reading Mercury, Oxford Gazette (Reading, Berkshire) of Monday 18^th September 1820:

To Vigil.

Sir,—In your strictures upon the late Reading Address, you talk of the Petitio Principii or begging the question, in other words, assuming as true the thing to be proved.
Pardon me, Sir, if I tax you in direct terms with being a most heinous transgressor in that way yourself.
In your aforesaid strictures you deal most abundantly in assertions unsupported by one tittle of proof.—Your trumpery quibbling on words I let pass at its full value.
In your comments upon the Queen’s Reply (meaning Answer) to the Reading Address you say,—“How much soever truth, justice and decency may have been violated in the late Reading Address to the Queen by the calumnies upon his Majesty’s Ministers and upon all the witnesses, heard or unheard, known or unknown, still. &c. &c.”
Now, Sir, I throw down my glove, or according to the more modern fancy, throw my hat in the ring, and challenge you to produce PROOFS from the Reading Address, of what you have thus upon two occasions so confidently advanced, and if I do not answer you satisfactorily let me be stigmatised as an enemy to TRUTH.
Reading, Sept. 11, 1820.

Another early figurative use of the phrase appeared in the preface to Blackwood’s Edinburgh Magazine (Edinburgh, Scotland) of June 1822:

Lord Byron¹, too, has written something about us—but whether a satire or an eulogy seems doubtful. The Noble Lord—great wits having short memories, and sometimes not very long judgments—has told the public and Mr. Murray that he has forgotten whether his letter is on or to the Editor of Blackwood’s Magazine. From this we fear his Lordship was in a state of civilation [= ?] when he penned it; and if ever he publishes it, as we scorn to take advantage of any man, we now give his Lordship and the public a solemn pledge, to drink one glass of Sherry, three of Champagne, two of Hock, ditto of Madeira, six of Old Port, and four-and-twenty of Claret, before we put pen to paper in reply. At the same time, Lord Byron should recollect that we are now an old man—just as Jeremy Bentham² is now an old woman; and that he, who has youth on his side, ought not to throw up his hat in the ring, and challenge us for a bellyful. We think we can fit him with the gloves, and that is pretty light play for one at our time of life. But we have still a blow or two left in us; and if a turn-up with the naked mauleys there must be, a hit on the jugular may peradventure do his Lordship’s business. Should his Lordship be dished in the ring—like Curtis or O’Leary—let the Reviewer who tries us remember that we wished to decline the contest.

¹ George Gordon Byron (1788-1824), English poet
² Jeremy Bentham (1748-1832), English philosopher and jurist