I finally managed to get a BSD on the additional Acer Aspire One (A150/ZG5) I got from a relative. There was once Win7 on the poor thing.
First I tried with OpenBSD, which I have on at least two other machines at
work (two running, but maybe there is another sitting around somewhere..),
but it looks like neither the Ethernet nor WiFi will work, and that renders
the effort somewhat useless. I then put FreeBSD (minimal version) on a USB
stick, booted, connected my LAN, and installed everything (so far except X,
because I intend to use the machine for text only stuff). That was over the
weekend, and in the meantime, I've also managed to start WiFi (can't say I
got it working, because I only needed to add some lines to loader.conf and
rc.conf), currently using my Android tablet as hotspot.
Update: now also working on my home wifi.
And yes, this I'm writing now on that FreeBSD Aspire, yay! So nice for a change, to have a complete system on 2 GB of disk space (95 GB left, will never fill it)!
Now it becomes reasonable to look for a spare/new battery, as the current one is quite dead. With FreeBSD working, it should be worth it, so that I can use this thing as a writing/coding machine.