I gotta assume the FA has no legal authority to arrest the girl

Granted, but the FA can someone with that authority.

[i]For what would she be arrested?

Theft of service. What service she stole is unknown. It's possible she had no valid ticket at all (faked boarding pass). It's possible she bought a less expensive ticket and tried to board a more expensive flight. At the very least, she was attempting to take an earlier flight and avoid a change fee. But again, if she had checked baggage, changing flights is not allowed.

after the airline rep allowed her to board

I reiterate, she snuck by the airline rep. She wasn't "allowed to board."