I would bypass the mother and deal directly with the boy.
Obviously this is a child you will want to closely monitor when your son plays with and does activities with, so you should have an instance pretty soon when you can do the following...
When you witness an outright lie get right down to his level, look him square in the eye and quietly say "You and I both know it didn't happen like that. You do not have to make up stories for me and (my son) to like you. We like you just the way you are. But if you continue to exagerate or lie it will make us not trust you. We can't be friends with someone we don't trust. Do you understand?"
The next time it happens repeat something similar and pretty soon he will get the drift that he doesn't need to lie or exagerate around you guys at least, perhaps it will rub off elsewhere.
It sounds like mom has to much on her plate to effectively parent this little guy. Sometimes it does "take a village" but the village has to be watching and willling to intervene when a situation occurs.
We had a similar problem with a boy one of my kids played with. I would just quietly and privately call him out on his lies and exagerations and pretty soon they stopped around us.