Now I turn on her phone, and it gets to the ROM's boot screen, and just as it's about to complete, it restarts again. I can enter TWRP (220.127.116.11) but it requires a password (which is a bug, afaik). I can press "cancel" and try to reset/wipe, but it won't mount or reset the cache partition, so it fails.
I used Odin (3.07, 3.09) to try and flash TWRP 18.104.22.168 and 22.214.171.124, and both failed. The result was Odin would reboot the phone, then it would enter the same bootloop, and the recovery would still be TWRP 126.96.36.199.
I tried downloading a stock S3 firmware and flashing it with Odin. Same problem.
I've tried new drivers. New cable. New USB port. I've tried using a different computer. Always the same result.
So a summary of what I can/can't do:
Can't use TWRP to reset/wipe
Can't use ADB Sideload
Can't modify device via ADB
CAN see the device via ADB
CAN enter download mode
I pulled DMESG at some point during all this. Maybe it can help shed some light: http://pastebin.com/4c2uL3Qq