Meanings
From the informal practice of putting ballots into a hat, and then drawing a single ballot from the hat at random to decide the winner. When one put one's own name on a ballot in the hat, one would therefore be eligible to win.
How to pronounce "put one's name in the hat":
AU
To run in an election or to nominate oneself for consideration in some other selection process; to nominate someone other than oneself for such consideration.