Sounds so much like that mother is insecure and jealous, what an obnoxious thing to say (the "showing off" comment)! Sadly there are people like that in the world, and they particularly target kids or adults who make them feel insecure.
One of the challenges our EG+ kids have in this world is they disproportionately attract attention of the insecure and jealous, and this can happen anywhere. IMO the worst is when it comes from a teacher in school (which I think happens more often than people realize, especially at schools with not enough at least HG gifted kids).
Maybe you can tell your son you don't think that mom is a very nice mother, and to stay away from her if possible. I suppose it depends on how much you have to see her whether you should tell your child to avoid her or not. I would not put it past her to say something negative directly to your child though, and I think preparing your child (just in case) by pointing out she is not nice could be a good idea. Then if she says something bad your child will just think, "my mom was right!" instead of taking it personally.
