It is distressing that your mom is holding on to this incident from two days ago. Is that her typical behavior in response to conflict? How long does it take her usually to return to baseline?

So long as she takes such a negative attitude to your friend, options other than hanging out at your house seem more appropriate. Perhaps your mother will become less judgmental of your friend over time. In the mean time, I wonder if your friend would be willing to drive both ways from time to time? And what are the chances of getting support in learning how to drive?