We understand passengers occasionally arrive at the airport without an ID, due to lost items or inadvertently leaving them at home. Not having an ID, does not necessarily mean a passenger won’t be allowed to fly. If passengers are willing to provide additional information, we have other means of substantiating someone’s identity, like using publicly available databases.
Passengers who are cleared through this process may be subject to additional screening. Passengers whose identity cannot be verified by TSA may not be allowed to enter the screening checkpoint or onto an airplane.
is from this web page.
In other words, yes, but they will need information that can be used to prove identity. They will need to have a web presence that can be looked up. Anything even resembling a photo ID, CostCO card, old high school ID, college ID, work ID's, especially anything with a photo can be used. The copy of the passport will not hurt. Once the TSA is satisfied that the substitute ID requirement is met, they may give the carry on a thorough screening, more than an xray. Your friend will likely get a thorough pat down as well.
In this case, tell the truth and plead for sympathy. This sort of problem happens all of the time and it is rare that it can not be worked around. Allow about an extra hour for the process.