Thanks to Robert's clear notes -
http://forum.tinycorelinux.net/index.php/topic,13827.90.html - I have successfully built extra modules for my mini-x, despite committing a newbie mistake on my first attempt. I tried to build modules without building the kernel first (ie. I didn't follow the instructions), assuming that the makefile would take care of any dependencies. I was mistaken - it appears that successful module building depends on some files which are created during the kernel build process. Attempting to load one of my bad modules gives this error:
modprobe: can't load module fuse (kernel/fs/fuse/fuse.ko): invalid module format
with this in dmesg:
fuse: no symbol version for module_layout
I was using the fuse module for testing because it was one I already had a good copy of, so could confirm that the good one could be loaded OK. Now that I have built a kernel, everything is fine - my modules work with the existing a10core kernel.
I did the build in an x86 TinyCore VM (running inside qemu on windows). Needed to load two more tcz's which weren't mentioned in the notes (but were pretty obvious from the errors) - compiletc and ncurses-dev. There's also a missing dependency for a "mkimage" command - this just means I didn't get the final kernel file out of it all. That's OK for now - I didn't want my own kernel. I imagine that grabbing mkimage from Debian would work.
Unfortunately for me, my module building has not been immediately productive - I have got my USB-serial adapter working, but it seems that the adapter doesn't work with my onewire interface anyway.
On an unrelated note, I'm now using sftp-server from the Debian Wheezy openssh package to handle sftp with dropbear. So, now I can ssh into the mini-x, and use sftp as well. I was getting tired of having to shuffle a flashdrive back and forth between machines.