Upvote:9
Your boyfriend will have to apply for a new passport at his local German mission in Mexico. The reason being that he is not looking to go back to Germany, and thus does not need a one-off document to get home. Indeed he wants to continue travelling and thus will need a brand new German passport. Once he obtains a new passport he will have to apply for a brand new ESTA since these are valid only for the passport they were issued for. Quoting from the linked Travel SE answer:
What should I do if the information on my passport has changed?
If you obtain a new passport or there is a change to your passport information, you must apply for a new travel authorization and pay the associated fees. See What information can I update? for a full listing of information that can be changed after submission of your application.
The application processing times for a new passport will depend on the mission, and on whether or not his case gets accepted as being an emergency. Once again: getting in contact with the German mission in Mexico should be the first step of this entire procedure.