Is there a way to measure the magnetic field of a digital device and mask it?
First up what they are actually measuring, this is mainly the DC current change in the power supply to the CPU, vs the CPU itself, as the CPU will have so many opposing field orientations internally, it is likely to cancel most of it,

Your looking for the highest current, in this case for a high end CPU, you might have 100+ Amps flowing through the motherboard, now this again should mostly cancel, but due to multiple grounds will couple some ratio of current through effectively 1 turn of wire, out of the power supply, down 1 loom, across the motherboard, and some ground current back to the power supply via other looms, its highly specific, so the exact magnitude of the signal would be hard to define.

For measuring it, the researches seemed to be using a phone magnetometer to listen to variations within a few Hz, you can attempt a similar effect

What everyday objects generate more magnetic field? that would be an electromagnet. in this exact example, it would be something with a wide frequency spread and random amplitude variations, but it would likely need to be some active circuitry as its the change in magnetic field that you need to make it very hard to recover, 

I suspect the amplitudes they are working with are similar to earths magnetic field for the phone to be a viable method, as such you can use something like a Merritt coil arrangement in your Faraday cage, and drive it to constantly shift between random orientations at random speeds. 

Edit: To expand a little, there are limits to how fast a phone magnetometer can sample (Datasheet), using this you can constrain the exact frequency range you need to cover, to make your jamming even more effective, the minimum is constrained slightly by practicality, the magnetic feild needs to change enough for the phone to recover it over the noise, so I expect it would not be a gentle ramp of current, but a 0-100% situation, so higher frequency content will likely be where jamming will hurt the most.