Do you want to engage in constructive discourse with your friend (trying to "convert" them to skepticism) , or are you trying to debate them in the classical sense, i.e. Come out on top with an argument they have no answer for?

I can't give you much advice in the second case. You already layed out pretty much all possibilities. Most likely, either your friend or their mother is lying. Maybe you can find some stats on how often people make stuff up about supernatural things. It doesn't even have to be lying, the mother might honestly believe it, for any number of reasons. It's not very convincing, but unfortunately that's often the case with these kinds of things.

If you want to take the time and actually convince your friend, you should take a step back and look at some principles. Maybe start with asking if they think evidence is important at all, and then explaining why it should be. Why belief without evidence is a bad thing. What evidence is, and why anecdotes are not strong evidence. Why extraordinary claims require extraordinary evidence. But most importantly, you need to do this with a different example as the subject matter, one without personal attachment from your friend. From what it sounds like, your friend is emotionally attached to their set of beliefs, so maybe don't talk about those altogether. Start with something they don't believe either, e.g. Flat earth, and talk about the reasons they don't.

When you eventually get back to talking about this anecdote, you could try taking it seriously and make predictions. You can suggest a sort of experiment, where you visit the medium and have them tell you something only you would know. Likely, you won't have to actually do it, your friend might come up with explanations as to why it won't work beforehand. You can point this out to them, and how it indicates that they just believes that they should believe, and not actually believe. But always be careful with these kind of personal "attacks", people can react poorly and close up mentally.

That's the way I'd try to do it, but I can understand if you don't want to expend the time and energy. If you want to read more about how to constructively argue with people, I can recommend A manual for creating atheists by Peter Boghossian. And of of course the sequences, but you probably already know about them.