It wouldn’t really have mattered if she had booked directly with the airline...after a very short time (usually 24 hours), they’d still be charging her for a name change. While it’s true that some airports might let her skate through with a different middle name to her passport, the possible expense if they do not would be horrific...basically they would charge her a full economy (if she’s lucky) fare, which could be three times the price she originally paid. So...I’d be urging her to cough up the $700, which morally seems a reasonable price for getting her own name wrong...I’d be chastising her for that, not for using a third party booking site!