You've probably done the following, so don't know if this will be helpful or not...
Scott had good advice about moving your guitar around- try positioning it at an angle to your monitor and see if any noise is reduced.
On top of the already excellent advice about using good quality cables, i'd ask you this: what kind of lighting do you have in your studio? Fluorescent lighting is notorious for adding noise to guitars, and dimmer switches of all types are evil; if you DO have lights on a dimmer switch, have your lights either all the way on or completely off.
another thing to check for, especially if you're having a pause every second or so, is to see if some sort of pump or motor is on the same circuit as where you're plugged into. Air conditioners and furnaces can put out stuff for your guitar to pick up, as well as water pumps and refrigerators.
I have EMG actives on my guitars, and they are pretty quiet for most things, and I know the frustration of noise hunting! My last suggestion would be to get a battery powered headphone amp of some sort (zoom makes little stomp box things, they can be battery powered and you can use headphones). Turn off everything in your studio, plug in the guitar to the headphone amp, pick an appropriately high gain setting comparable to what you've been using on the computer, and listen for the noise. If you hear noise, you can be assured that it's not the computer, and is probably some strong EMF in the area (like the airconditioner, etc but cell phone towers and cable tv towers can do this if you're close enough. Same with power lines passing overhead). If you do NOT get noise, turn on your computer while still "off the grid" with the headphone amp. If you have noise, you'll know it's something in the computer.
many ideas, most probably useless...