1GB RAM is the bare minimum for Snow, and if there's a problem with that it might be causing this issue. Snow Leopard may be more demanding of resources than Leopard was and expose pre-existing problems that weren't noticeable before.
I would definitely run the Hardware Test in Extended mode, which, among other things, will do at least a superficial check of the RAM. And it may turn up something else besides RAM.
http://support.apple.com/kb/HT1883
If you want to test the RAM more thoroughly then
To test the memory, get memtest and run it in single user mode, where it will test as much memory as possible, more than with the OS loaded.
You can get memtest + directions from the link below. However, ignore running it from Terminal. Instead, boot into SU Mode, Cmd-s at the startup chime. (Best to startup from a full shutdown.)
At the prompt, simply type /usr/bin/memtest all 3 -L (From this link It will be installed in /usr/bin/) Then hit return. This will run three loops of memtest and create a log in Console in Utilities.
If you want to run memtest longer, which may be advisable, since RAM errors can be very elusive, just remove the "all 3" which will give you /usr/bin/memtest -L and hit return. It will keep testing until you quit it.
If you want to quit the test, just hit control-c
When finished, you can just type in "reboot" and hit return.
http://osxdaily.com/2011/05/03/memtest-mac-ram-test/
Direct link for the download.
http://cdn.command-tab.com/2008/memtest_422.zip
I would also run SMART Utility, linked above, since the native SMART Status check for the drive isn't that thorough or reliable.
