Throw one’s hat into the ring

Means to make or take up a challenge or demonstrate one’s willingness to join an enterprise and dates in this sense from the latter half of the 19th century. The ring referred to is the boxing ring, which these days is a square, but in the early days of prize-fighting it was just an open space ringed with a crowd of onlookers. In those days, a pugilist would sometimes issue a challenge to fight anyone from the crowd and anyone accepting such a challenge would typically throw their hat into the ring.