http://www.linuxmint.com/edition.php?id=143
Linux Mint 16 32-bit versions
Cinnamon is 1.1 GB
NATE desktop is 1.2 GB
KDE desktop is 1.3 GB
Xfce desktop is 1.1 GB
I'm wading through the Mint forum right now. Have found so far that you need 512k to run Mint13/Cinnamon and the download iso is 817 meg. Think I'll start there and see if it runs off DVD, and work up. With the chunk that the video on that laptop takes out of the available ram, I'm at around 948 meg available out of the 1 gig. Also checked and according to HP, 1 gig is maxed out on that unit so no option to add more. Downloading the iso 'as we speak'... ;-)
A little further digging and I find that Mint versions above Mint13 require a PAE enabled processor and that if you try to run it on a cpu without PAE, you will get a flag (which I didn't get.) Supposedly AMD Athlons are PAE enabled so that shouldn't be an issue here. Mint16, 15, and 14 also need 512k (1g optimal) to work well if installed. I am not seeing anywhere (yet) about what is needed to run it off DVD, but I would bet on that upper end figure.
In thinking back, running out of overhead seems to be what happens, because the display goes bonkers, tending to indicate that the ram is getting overwritten, and I would bet that to run it live it wants to load all the libs into ram. If one of the smaller footprint versions runs off the liveDVD, it will prove the point. Mint 13 has the advantage of being the LTS version (but I kind of like the improvements they made in the desktop with v16..) It does seem that any of these versions should run ok if I do an install because all the library stuff will be on hard disk instead of in ram, but I want to get something running off the DVD first to see what issues with drivers and such there might be before I commit to an install..;-)