If my cousin totaled my car, but the insurance is in her name, will her insurance cover the damages to my car?

A car insurance policy follows the vehicle rather than the driver. This means that her insurance policy only covers her vehicle, not yours or anyone else's vehicle she happens to borrow. In order to have your vehicle's damage covered you need to have your own insurance policy on your car and have collision coverage.
If you did have your vehicle insured and lent it to your cousin then she would have been covered to drive the vehicle the same as if you were driving it. However, if you lend your car and the driver has an at-fault accident while driving your car than your rates are going to go up the same as if you yourself had caused the car accident.

If you have insurance on your car you can let anyone borrow it that doesn't live in your household and they are covered. That they have insurance on their own car isn't going to cover your car if you do not have it insured.