It looks like there is an I/O Error (STATUS_NO_SUCH_DEVICE, the second 0x0000000E) when Windows tries to access the pagefile. Looks like a defective harddisc to me. You could try to run an error scan (using HDtune or any other HD scan software). It may also be a defective HD controller so it would be worthwhile to scan the HD on another system.
Do you see anything in the event logs (Event Source should be 'Disk')?
I've to admit it is a bit strange that it only happens when you try to access internet.
It's not when she access the internet, it's when there's a live internet connection. An ethernet cable gets plugged in and in 10 minutes the BSOD comes up. A regular phone cable gets plugged in and in 10 minutes the BSOD comes up. I gave up...told her to get a new motherboard >.<