This NetGear device looks like it would fill the bill nicely, and if the info on that page is to be believed, it connects to a router to allow any computer on the home network to access the data on it.
Problem: $230 before you put any drives in it.
Still, adding a NAS box seems to be the way to go, here. The thing requires minimal configuration and administration; basically you pop drives in, set the thing up, and then dump data to it. If a drive fails, remove it and put in a new one; it rebuilds automatically, and some of these things even let you hot-swap them.
Around $500 would seem to get you about 1.5 TB of reasonably reliable network storage. That is, at present, a lot; and as drives get bigger I could simply swap in larger drives.
Doing this would also require making a few changes to the configuration of the home network, though. I'd want to change where the modem plugs in; I'd also want to get a UPS solely for the modem, router, and NAS. This would mean either running ethernet cable to my Mom's computer, or upgrading her computer to something a little more modern, argh etc. (Mom's computer is a vintage 1998 Compaq Presario desktop. She doesn't want a new one.)
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I want to hit the guy in this picture with a tire iron.
...really, that's all that needs to be said about this.