> Why would anyone value this randomly invented money?
Because they can use it to pay for goods?
Your notion is that almost everyone is going to be out of a job and thus have nothing. Okay, so I'm one of those people and I need this house built. But I'm not making any money because of AI or whatever. Maybe someone else needs someone to drive their aging relative around and they're a good builder.
If
1. neither of those people have jobs or income because of AI
2. AI isn't provisioning services for basically free,
then it makes sense for them to do an exchange of labor - even with AI (if that AI is not providing services to everyone). The original reason for having money and exchanging it still exists.