Sounds to me that you got one with a "Loose" CT [Current Transformer]. The CT is obviously vibrating against it's own mounting means.
Could also be caused from "Noise" on the AC system. That would make a loosely wound or loosely mounted CT hum, along with possibly having humming sounds created by the control circuitry [Diac, trigger, SCR, etc.].
If the sound is faint [barely audible] and the device tests out properly, it's most likely just "Nuisance Noise" - a result of cheap design or sloppy assembly by the manufacturer.
If the sound is loud [easilly audible] and the device has difficulty test tripping, I would suspect the triggering SCR has a few too many charges passing from Cathode through the Gate [semi-shorted SCR].
In that case, remove the GFCI, place it in the Recycle bin, then install a new GFCI
I have noticed a few GFCI Receptacles which hummed slightly [<5dB], and some that the vibrations can be felt when placing your palm across it. They functioned properly when tested, using various techniques [external loads and/or test button]. These were on Commercial projects with Conduit systems [metallic boxes, conduits, cables and raceways] and Steel Studs, so I figured the sound was more prevailant with all the metal to vibrate through.
As mentioned by others, it wouldn't hurt to replace the device if you do not feel comfortable about the audible humming.
P.S. Is this a GFCI Receptacle, or GFCI Breaker??? If it's a GFCI Breaker, that humming could be the normal contact vibrations of the breaker frame, the trip spring inside the breaker's frame, or the contact points inside the breaker's frame.
Some of these symptoms are normal and not an immediate danger. Some of them can be signs of hidden trouble.
If the humming increases linearally with load, but dissappears with no load, check the mounting and Line [bus] contacting points. Tap the breaker lightly with the back of a screwdriver's handle just in front and behind the handle [on the upward facing surface] to see if the humming varies.
Check the voltage drop across the breaker at various levels of hum. If the VD changes with humming, suspect poor contact seating inside the frame.
BTW: The contact seating pressure situation could be another source of hum in the GFCI Receptacle! Once again, checking the voltage drop across the device [both "Hot" and "Neutral"] with some light tapping can verify this one.
The problem is you need to work with a Live circuit and exposed Receptacle terminals. BE VERY CAREFULL!!!
If you are by any means unfamiliar to working live circuits and techniques to isolate your body from ground / grounded parts of the structure, PLEASE DO NOT ATTEMPT THESE TESTS!!!
Just purchase a new device and replace the noisy one. The $10 - $20 spent is absolutely nothing compared to your life and well being!!!