I settled on a Dreamplug computer, running Debian Squeeze on an ARM processor, with a 10" USB touchscreen monitor. The choice of hardware limited the available software.
I wanted a computer that is as small as possible, so that Mum would feel comfortable with it. I wanted a Linux box. I use Windows at work; at home I use only Linux. I believe it's better suited to this type of thing than Windows. And of course I lack skills in Windows to do what I want. That's probably not the case for many people.
I also wanted a touch screen, just in case Mum or a friend or neighbour ever need to control the computer. I learnt that touchscreens are not well supported on Linux. I considered Android or iPad tablets, but these don't allow VNC connections. The USB touchscreen requires a stylus (supplied).
The Dreamplug came with Linux preloaded, and started up first time. No keyboard or mouse is supplied. I used the ssh (secure shell) program to configure the Dreamplug from my computer. I tried to use one of the soft keyboards that are available, but it was just too painful.
I used a couple of spare Homeplugs to connect the computer to the modem/router. A combined microphone, speaker, and webcam completed the hardware.The Dreamplug is the orange and black box plugged in to the extension lead.