If there is a cosigner, then bankruptcy on the part of your girlfriend won't be helpful: the mother will simply be fully responsible.
The first option here should be to talk to the dealership/lessor. It's entirely possible they will be willing to work with your girlfriend in order to avoid a default. Lenders don't want defaults, they don't want to sell off your account for $0.20 on the dollar or anything of that nature. So they're often willing to make a deal - either reducing the total amount owed, or allowing payments over time. But be up front about everything immediately. Be prepared to prove it was stolen (police report, etc.); and understand that the lack of insurance probably voids the lease. That's a sunk cost though; don't try to hide any of it, just be honest and forthright and ask nicely.
Second, your girlfriend could try to obtain a personal loan (unsecured) to pay for the difference. I don't know her credit, but if it's decent it's possible to get several thousand dollars at a non-usurious rate (10-15%). It's at least worth a shot. If her credit is in the "gets random credit card offers in the mail that don't have annual fees" range, she probably qualifies for something like this. But if she's going into a lease with a cosigner, perhaps she doesn't have any/good credit - but perhaps a few years later she does now?
Third, your girlfriend should talk to her mother. She should do this anyway; the first option will possibly have a negative impact on her credit as well. It's critical to make sure she's open with her mother as well: hiding this won't be possible, and not telling her about it and letting her find out from her credit report is a big mistake.
But if the first two options don't pan out, she would have to involve her mother in some way; assuming the mother can either pay it off or get a loan herself, anyway. A reasonable solution here might be for the mother to get the personal loan and have your girlfriend pay that personal loan, though I recommend in that case that she pay her mother rather than pay the lender (so her mother can pay the lender and ensure no missed payments).
Finally: don't get another car lease, unless and until you two are financially comfortable and independent. This was a mistake going in; buying something that requires you to have a cosigner means you're buying something over your means.