@barrowman: If you're versed in Linux for that long, this should be a piece of cake.
First, download the ISO you think will fit the purpose of the netbook.
Burn the ISO image onto CD or USB depending on which netbook you have.
Update your BIOS settings to set the device to boot from the new CD or USB drive.
Once you've booted the CD/USB device, you'll land either at the shell or the TinyCore desktop.
Since "an old netbook" tells me nothing about the unit, whether it's wifi capable, ethernet capable, 32 or 64 bit, how much RAM, etc. I'm blind as to what's working out-of-the-box when you boot TinyCore so you'll have a little hunting to do in order to figure out your hardware specs.
If you HAVE internet access at this point, your next task would be to install tc-install to assist you in creating a bootable version of TinyCore where Mint used to live.