In order to proceed with a case against the prison system, she would need to be able to show "deliberate indifference" which is a higher standard of care than mere negligence. She (or you) should talk to a lawyer who specializes in so-called "1983" actions, specifically, the 8th amendment. I can tell you that a slip and fall at a prison is likely a no-go.
Mr. Hoffman is right, these cases are incredibly complex and it will take at least one (and possibly several) attorney appointments before you know if this is worth pursuing. Your attorney will need your mother's complete medical records and history, but that doesn't mean you should wait to see someone until you have those in hand. Make an appointment tomorrow and see a few attorneys asap.
Right now, the best thing you can do is inform your personal insurance as well as the insurance on the rental car, preferably in writing. If the damage is minor, you might want to work out an agreement to pay for some or all of the damage. You would want a written agreement saying you agree to pay x amount and he agrees not bring any suit or other action against you in the future. Either way, a quick phone call or consultation with a lawyer will likely be helpful.