Probably a stolen card from their last airbnb rental/trash/steal creditcards.

Or a prepaid debit card (do those work with airbnb)?

The interesting thing will be the gmail account's IP history. I suspect SFPD will get the IP history and I hope this DJ Pattrson logged in from an identifiable location (or that the phone number, presumably a prepaid, is linkable to location and identity somehow).

SFPD is one of the better police forces to have to work with on this; I'd sure rather be dealing with them than Oakland or various other East Bay police forces. You might also want to involve REACT (http://www.reacttf.org/), the Silicon Valley (incl SF) high tech crime task force. They'll be good at tracking someone down through online identity, compared to local police.

I wonder if AirBnB has offered the host/victim some AirBnB credit to stay elsewhere while her place is being de-trashed.

Well the solution there is not to accept a credit card as a form of identification.

I check into lots of hotels and about 50% of the time they require an actual government-issued photo ID; I assume for room-trashing situations and the probably more-common situation where somebody racks up a huge bill and then steals away in the middle of the night leaving nothing but a stolen CC number.

It's a lot more difficult, and the legal penalties are much steeper, for forging a government-issued identity document than for using a prepaid credit card with a false name on it.

You're giving thieves and vandals far too much credit. For most, if they had the intelligence to get away with the crime, then they'd have the intelligence to know not to commit it.

I hate to break it to you, but people are this stupid and do try to use stolen credit cards for hotels all the time.