please note that we both have auto insurance. so technically, who's auto insurance rate would increase due to the accident - friend's insurance who was driving my car at the time of the accident or my insurance since the car was lent to her?
Yes it can if it was your friends fault. If her fault and your car, both of your insurances can increase. If not her fault and your insurance company doesn't have to pay out, then probably neither of your insurances will increase.
The above is general information only and not specfiic advice since there are not enough facts.
You need to speak to your respective insurance companies about rates. This is not within the realm of what personal injury attorney's give legal advice on.
If you are asking about fault or potential liability for a claim that is a different issue.