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Does "next player's turn" mean "next turn of the player" or "turn of the next player"? Or can it mean either depending on the context? In this particular case the context is a two player board game such as chess (but not chess) where players alternate to take their turns.
Meta: also I'd appreciate some help with the good question title, as mine clearly is not good.
It means that it is the other player's turn... because you have finished your turn.
e.g. In chess, you move your Rook (a chess piece). So, your turn is over. It is now the next player's turn, so you must wait for him (her) to finish.
In light of comments to the question, I think we have to accept that syntactically, constructions of the form the [adjective] [possessive] [noun] are inherently ambiguous. Consider, for example,...
Putting aside the fact that this is informal British term for a weak, cowardly, or oversensitive man, there's structural ambiguity as to whether the "literal" meaning is a blouse belonging to a big girl, or a big blouse belonging to a girl.
With OP's example, we could in principle contrive a context where a game involved turns taken by both players and the game controller - a betting game, for example, distinguishing between (any) player's turn and the banker's turn. Feasibly therefore, one might say "We'll take a break after the next player's turn" (but there might be a banker's turn before that).
To make the context even more contrived, if we suppose there's only one player in the game (along with the banker), he could be identified as the player. It would be meaningless to speak of the next player, since there's only one, but we could validly refer to the next turn of the player.
TL;DR: By default, next modifies the first "credible" NP after it. It would be semantically perverse to interpret player's turn as the likely referent, given that player is perfectly credible. The possessive apostrophe modifies the nearest credible NP before it (in OP's case, that's next player).
