All RAM gets old and sometimes a little temperamental. But yeah, follow the above advice: MEMTEST to check that the RAM is fine, then disk diagnostics incase the HDD is going bad. You might also want to try taking the HDD completely out, wiping any dust off it, then putting it back (reseating to clear any static), then do the same for the RAM. It might help.
And as I mentioned in another thread, look in the BIOS settings to see if there's a setting for "Halt on Errors" or words to that effect. There might be a similar setting in MSCONFIG but I'm not sure. Anyway, it should stop the laptop from resetting automatically on BSoD, allowing you to at least write down whatever information it gives you.