It is possible to get a passport renewed very quickly, however the fastest service is reserved for life or death emergencies, for which I would not imagine a holiday to Cancun can count. Getting the passport renewed will require your son to make an in person (required for persons under 16 anyway) visit to a passport agency, having already made an appointment.

According to the US Bureau of Consular AFfairs:

Expedited Service at a Passport Agency

If you need a passport very quickly, the best method may be to apply at a U.S. Department of State Passport Agency.

There are 28 passport agencies around the country that can provide expedited service. Many have opened in the last few years and there may be one closer to you than you think. Appointments and proof of immediate international travel are required to apply at U.S. Department of State passport agencies. For a complete list of agencies, click here.

To schedule an appointment, please call the National Passport Information Center at 1-877-487-2778; 1-888-874-7793 (TDD/TTY). The $60 expedite fee is required for each application submitted in person at a Passport Agency or Center. Payment methods include:

• Major Credit Card – Visa, MasterCard, American Express Discover
• Debit/Check cards (not ATM cards)
• Checks, bank drafts, money orders
• Cash payments must be exact change. No change is given.

The passport agency will determine how best to return the passport in order to meet the customer's documented travel plans. Will Call service is offered for life/death emergencies and immediate travel. Proof of immediate international travel is required for will call service.

The timescale they normally give for these is < 8 days, so I'd say you'll be doing very well to get a passport in the 24 hours you ask for (although user @Aleks G indicates that an acquanintance of theirs was able to get one in this timescale). Still, call the number and give it a try (if you can travel to one of the agencies)