It turns out no one got to keep the stick after all.
From the New York Daily News:
...ask the Rangers fans who fought for the King’s stick when he threw it into the Garden stands after being named the game’s first star. A minor skirmish broke out in the crowd when no one would let go, prompting ushers to break it up and take the stick away. No one would get the biggest souvenir on this night.
UPDATE: Watch the stick be taken away by security in the video below: