So your mom was not denied entry by Taiwan, she was denied boarding by an airline. Airlines are paranoid about these things because they get fined and have to pay the trip back if they bring in people with improper documentation, but for things like borderline passport expiration you can usually get away it by signing a waiver (yes, I've done this). But if you get a really recalcitrant check-in agent about all you can do is try to reach their supervisor and/or get the destination country's embassy on the phone.