Hrishi's right, it does indeed stand for New York Stax Exchange.
It was, very briefly, a team name for myself and a few others, before it became the moniker for rebooted Vintage tournament series that I started running again in 2009. The N.Y.S.E. Challenge was a monthly Vintage series that I ran out at Brothers Grim. I ran 20+ of those, then abandoned the name when Nick Coss and I started running the N.E.V. (North East Vintage) series after that. I stopped running monthly Vintage events after the N.E.V. Invitational, and started running the N.Y.S.E. Open annually after that (though I did skip 2018).
With every NYSE Open that's run, a limited number of qualifier tournament are run. For the price of an entry fee, a T/O is able to run an event guaranteeing the winner of the qualifier free entry to that year's NYSE Open.
I currently have seven qualifier spots left open. If you'd like to run a qualifier, please reach out to me and let me know!