If the ATI driver in Xorg doesn't work for that chipset, you might want to try the Xvesa server instead. Hopefully, the chipset isn't so old that it predates VESA standards! :)
If you want something in the Ubuntu family, Lubuntu might be a good choice for something ligher-weight than Mint or Ubuntu. (It uses the LXDE desktop.)
LXDE on Ubuntu (Lubuntu) and Debian is a great way to save precious RAM, still get a decent desktop interface (graphical panel, menu, file browser), and still have access to the best repositories of well maintained software from a major distro. I even use machines with decent RAM as I really don't need more functionality than it provides and prefer to leave RAM available for caches etc.
Plus, with no OpenGL required, it works under vnc, so I'm able to stay consistent remote and local.
XFCE has the same qualities but uses more RAM.
On ubuntu 12.04 the ati video driver package is xserver-xorg-video-ati. Remove that (and xserver-xorg-video-all which depends on it) and see if X falls back to using xserver-xorg-video-vesa .
Video playback and Adobe Flash support probably will be bad to non-existent though. (And old Adobe Flash plugin for Firefox is a dead end anyway in terms of future security support, the Adobe Flash support baked into Google Chrome is going to continue to be supported, but it requires OpenGL support)
In terms of desktop environments, definitely don't mess with anything that requires OpenGL like Gnome 3 or Ubuntu Unity (they have a 2D version of Unity that they're going to abandon, it always sucked anyway).