He's saying that he doesn't believe the contention that she is on the no-fly list. The government denies that she's on a no-fly list. The document that claims she was comes from an airline, not from the DHS itself, so it's possible that the reason she was denied boarding comes from further down the line.
She's reluctant to try again, since the flight isn't cheap (and they didn't refund her money). The airline is blaming DHS, and that's the part I'm not sure how they'd go about proving. They'd need to prove that the order they claim came from the DHS actually came from the DHS. I don't know what channel the message was delivered to them, so I don't know how they'd authenticate it, and the fact that DHS usually operates in secrecy makes it that much harder.
If I read it correctly, she doesn't have to try again, she simply needs to get Malaysia Airlines to cough up their source for the document they provided. I've got no idea how easy that would be.