As all the cool kids have stopped using these little machines since they bought their shiny new tablets, there are a lot of them cheap on eBay. It struck me that one of these with a light Linux distro, such as Peppermint would be a handy thing to have on the bedside table as a wifi radio, music streamer, email and web browser, etc, without taking up much space. Does this idea make any sense?
Yeh, great idea … personally I have a few Acer Aspire Ones, and a n MSI Wind for similar tasks 
Definitely.
You can do far more with that machine. 
I have rescued one of these from being scrapped (EEPC 1000H) with 160 GiB HDD and it works like a charm.
I absolutely LOVE my netbooks, and wouldn’t be without them now.
Though I do think netbooks aren’t netbooks without an SSD … you can be so much rougher with them with an SSD