A brief summary: I have the same problem, and I do not have the answer. I want to dispute the accepted answer, as a means of constructive debate, as well as provide more information in order to get to the bottom of it.
Just as you stated, the noise comes from anything going on on the screen, such as a cursor blinking. The noise is audible when both mouse and keyboard are not in use (noise persists if the peripherals are switched off). So let's put this hypothesis to rest (switching out mouse/kbd for wired). The noise comes from either the video card or the monitor.
Now, to the quality of headphones. I have a somewhat budget audiophile setup (by hi-end audio standards) that consists of separate components: DAC, preamp, power amp, quality headphones, quality cables (USB, speaker cables, interconnects). Let's just say that each component is no less than 1000$, with some costing way more. So the noise is not due to lack of insulation/shielding in the headphones or any other components.
That leaves us with the sound card (a dedicated DAC in my case). The noise originates somewhere in the video card, then travels via the USB cable, and makes it into the analog stage, despite the assumption that a good quality sound card should filter out any noise.
To test this hypothesis, I am going to do an experiment: connect a digital oscilloscope to various points of signal path, from the USB cable to DAC output. Without any music being played, the "cursor noise" should be clearly discerned.
I will update this post in a few days. In the meantime, I suggest you un-accept the currently accepted answer in order to keep it open and encourage potentially knowledgeable people to post answers.
The scope was not necessary. Upon discussing the issue with the DAC vendor, it was suggested that I might have a ground loop. Indeed, I reshuffled my cables following renovation of my living room, and ended up with a laptop and a DAC plugged into different wall outlets. As soon as I moved them to a common power strip, all noise disappeared completely.