Seems like this is sort of like "losing" the return part of your tourist visa. We had a friend visiting last year - only about 10 days. So she had not overstayed her visa but did leave the return portion at our house when we took her to the airport. She simply went to the INM window and told them she had lost her documentation. They didn't ask any questions at all - simply filled out the return portion of a new FMM and handed it to her, Took less than 10 minutes. If I remember correctly the fee/fine was $306 pesos.
Depending on the airline you should probably go to the INM window before even trying to check in with the airline. Many of them will not process your check-in until you can give them the return portion of your visa.
Thanks for helping out your friend. This shouldn't be too complicated or expensive.