Why is the GFCI buzzing when you turn on the Breaker?

There's a few small parts inside a GFCI and also a small coil of cable. Electricity in an AC circuit can be considered to vibrate in the cables, this produces unnoticeable harmonics (vibrations

/noise). However when a coil (cable

wrapped around and around) has AC Current flowing through it these harmonics are increased and become noticeable (can

be heard).

The noise coming out of your device is not a safety issue, it is not indicative of a fault with your electricity, however it is annoying.
I shall now give a couple of reasons why it may be unavoidable.
Resonance depends on many things... Materials the board and GFCI are mounted to being one. the overall measurements and position of the GFCI in relation to the materials around it being another. It might be that the board itself is spreading the sound wave.
It may even be the tiniest spec of dust inside the GFCI itself, or that the GFCI's

mounting bracket is a tiny bit loose.
You may find you can slide The GFCI to one side or another, space permitting, inside your board (you

may have to remove a black plate from the front panel), and by just moving it to one side it may stop the noise all together.

A rather unorthodox method of stopping this noise (due

to dust and other internal causes), is to give the GFCI a sharp rap on the front with a solid Item not a heavy item example:

Don't use a hammer, also try not to hit the switches). something like a spoon or the handle of a screw driver should do the trick.
(Source: Qualified electrician)

Otherwise, your looking at maybe calling out an electrician to see if he can find where the sound is specifically coming from, and to put a piece of dampening material into your board.

If in doubt, get an electrician out.