Well that's absolutely horrible, SO sorry!
This happened to a client of mine, they were not allowed to leave Mexico and had to go to the US Consulate in Cancun to get a replacement passport. It did take a few days, the airline was called and their flight schedule changed (yes they paid change fees and additional airfare
). They had to get a photo and pay a new passport fee. They did make a copy of their passport and had it stashed in their luggage (thank goodness) so it was relatively easy to verify their personal info and issue a new passport so no other ID was necessary for them.
*edit* I believe they went to Cancun because their hotel was closer to Cancun than Playa. But I'm not sure if that was the reason why, or if they HAD to go to Cancun. Also, call your airline so they don't put you down as a no show. If you don't show up for check in, the ticket is suspended and has no value.