Can't say for sure what the problem is, but it might be the graphics drivers, or it could possibly be flaky hardware.

First I would check the BIOS settings, if it is overclocked at all, return it to default speeds and voltages. Also try changing the amount of RAM the integrated graphics uses to min and max if you can, to see if that has any effect.

Next try a different version of Ubuntu (e.g. 10.04 which is the LTS (long term support) version).

You might also want the check for bad RAM, set the RAM for the graphics to the minimum and run memtest86 (which is installed by default with Ubuntu you just need to select it from the GRUB boot menu), you'll want to leave this running overnight at least, maybe 24 hours if you can, this is likely to pick up bad RAM but not guaranteed to, it has missed bad RAM I had. I don't think the problem is likely to be caused by bad RAM, but it is worth checking anyway especially since this is an old machine.

Failing that you might want to pick up a cheap graphics card to replace the internal graphics. I think NVidia is the best choice here, but I haven't looked at graphics cards for ages so you might want to do your own research on it.