Agreed on the ChromeBook.
I'm not sure about making a ChromeBook use dial-up, so the solution is to somehow get a WiFi router on dialup.
I think there used to be WiFi routers that could manage a modem directly, but there isn't much call for them these days so I doubt you can find one.
You could set up a computer with Linux just to manage the dialup, and plug that into the router's WAN port. But maybe you can just customize a router to do what you need:
Buy a router that is well supported by open firmware and has USB ports. Install the open firmware, login as root, then customize the router to do the dialup with a USB modem.
In the past, I have used TomatoUSB with an Asus RT-N16 router (costs about $80 new). It was a pleasure to work with. The router gives you about 24 MB of usable storage using onboard flash memory, but you can trivially plug in a USB flash drive and have gigabytes of storage if you need it. But you can probably set up the needed scripts to manage the modem in the 24 MB space.
There are newer routers with bigger onboard flash if you prefer. I only mention the Asus RT-N16 because I have actually worked with one, and it's very inexpensive. And it has plenty of CPU speed and RAM for this application.
The above solution is cheaper than using a computer to manage the dialup, and should be bulletproof. Also your relatives are unlikely to mess with it.
P.S. Hmm, I did a quick Google search and there are still routers with dialup support. Here's one for about $150... I've never used one so I don't know how well it works.