So sort of like a 'temporary url' some providers offer?
I'm going to file this under 'Don't do sketchy shit' category, especially if this is a shared hosting server.
No doubt it can be done with a janky enough unsupported config of your webserver which will break at 11pm on a Friday.
An easier way is for them to think up a domain name, and add that to their hostfile. You will have to stick with self signed SSL. Sticky the shit out of their customer profile on what they are doing when it comes to debugging when they get a new computer.
If this is public facing, tell them to stop being a cheap ass and buy a domain.
