I would say your best bet is to leave via an airport where US citizens can use the Biometric/eGates. This minimizes the odds of scrutiny by border officials. Make sure you buy a direct flight from said airport to the US, in case you do end up facing trouble, as this would minimize the odds of being delayed. Currently such eGates are available in:
- Belgium
- Finland
- France
- Italy
- Netherlands
Another option is to exit on foot via the Spain/Gibraltar border. As outlined in How does Schengen know I didn't overstay if exiting through Gibraltar by air?, the exit checks conducted by Spanish authorities at that location are often lax, especially on the weekends. Even if a stringent exit check is conducted, there's a chance that the border guards will be lenient to your friend and avoid imposing an entry ban, as there's generally more forgiveness towards citizens of countries that don't require a visa to enter the Schengen area. Make sure you're extremely polite to the border authorities and apologize for staying so long. Make sure you have a ticket out of Gibraltar to the US before crossing the border, in case the UK border officials notice your overstay. Having a ticket and stating that you're only passing in transit should alleviate their concerns.
Good luck and please don't overstay in the future - the consequences can be extremely unpleasant and the extra days are never worth the legal hassles.