If my daughter caused less than $1,000 in damage to another car, should we still file a claim or pay out of pocket?

My 20 yr old daughter hit someone and tore the owner's rear bumper cover. The cost of replacing the cover and painting is $854 at a local body shop. At her age, it seems that it's likely that the future premium penalty will be greater than the cost of repair.

I would agree with your assumption. In Kentucky, an at-fault accident impacts rates by an average of $917 per year (page 58 of the report). Since the accident will impact her rate for 3 years, that's quite a bit more than the cost of the repair. I would recommend paying for the other driver's damage out of pocket if it's possible.