Sorry @Susann20 but this is down to the guest not Airbnb (and in some part you for knowingly accepting a third party booking contrary to Airbnb's T&Cs). If the son wanted to pay for the second month, all he had to do is send his mother the money. And the payment could have come out of her visa account.
There is absolutely no way the son could have paid Airbnb, as he is suggesting, unless his mother gave him access to her account and he changed the payment method. In which case the payment would have gone through to Airbnb without an issue. Is that what he has shown you proof of?
By the way why did you accept a third party booking for a long term stay rather than asking the son to set up his own account and book directly?
You are right if he doesn't sort out the payment with airbnb yu will need to ask him to leave.