There are two ways, and it'll depend on how long he has in the US.
1) He can apply for a new passport, from America, through the Consulate. For example, the Consulate of Panama in Houston, Texas states that it's possible to do so and how to do it, including the requisite forms and costs. The downside is that it takes 6-8 weeks, which may be too long for him.
This is confirmed on the pages of the Consulate in Toronto, Canada, the Consulate in Los Angeles and even the one in the UK suggests 10 weeks!
2) The second option is to, according to the UK consulate 'apply for a travel document'. This is vague, but I'd take it to be like an Emergency Travel Document that New Zealand has for citizens who have lost their passports overseas - it's like a pseudo-passport, generally only available for a short trip (usually back home).
For this I'd recommend trying the consulate again, or contacting the Panamanian Passport Office back in Panama for information on this 'travel document'.
These are recognised by most countries, including the one he's in - the Department of Homeland Security insists you have either a valid passport, or 'an emergency travel document issued by an embassy or consulate.'