Normally speeding tickets are written to a person, the driver of a vehicle, and not the vehicle. Due to how the citation is written your friend's driving record will be the one that should show the conviction for the speeding violation.

Once your friend pays the ticket the violation should appear on her driving history and thus her insurance rates (not your mother's) could be affected depending upon her insurance company's rating system. If your friend takes care of the ticket properly it is doubtful that your mother would find out about the speeding ticket that someone received in her vehicle. You can check with the NYS Department of Motor Vehicles to be certain.