MacOS – Time Capsule partly stops working after some hours

Run iStumbler from your Mac. (Get it here). Look at the other wireless networks in your vicinity - and check what channel(s) they're running on. You want your device to be as far away from those channels as possible. There is a 3-channel overlap. That means, a device on channel 1 will interfere (albeit only slightly,) with a device on channel 3 (or 4? I forget.) So if lots of devices are on channel 6 (most come out of the box on this channel,) you want to be on 9/10/11 or 3/2/1. I almost always just use 1 or 11.

There is also a "use network robustness" option which I think increases range but decreases bandwidth. See this Apple Article.

You don't need OS X Server for Time Machine backups. Regular OS X works fine so he could just repurpose the white MacBook into a backup server with the external hard drive and get her an Air instead.

I agree that he should confirm that he doesn't have network issues with both his wireless LAN and WAN. Losing the Internet connection is more likely to be a modem or provider issue than a TC issue. Switching the connection to another router can confirm it. The new Apple AE is a good choice but an Extreme might be better if he needs multiple Ethernet ports at the router location.