It could be your brake booster. The brake booster is the device that puts the "power" in power brakes. It operates from engine vacuum. A momentary sigh from the brake booster is harmless.
To see if it's the brake booster, sit in the car with the engine OFF and step on the brake several times in a row. You'll notice that the brake pedal offers more and more resistance with each press. When it doesn't seem to be getting any harder with each new press, depress the brake once, then release. Doggie noise or no noise? Assuming no noise, now start the car and let the engine settle for a few moments, then depress the brake once and release. Doggie noise or no noise? If you got no noise the first time -- after exhausting the vacuum in the brake booster -- and a noise the second time, that's pretty convincing evidence that the brake booster is making the noise.
There's no reason to remedy this unless you can't stand the noise. It's not a sign of trouble. Some brake boosters sigh, that's just the way it is. I can't explain why it would stop sighing when it rains.
On your brake calipers there are slider pins and there are ways (channels) in which the brake pads move ever-so-slightly when the brake pedal is depressed and released. These are normally inspected and/or lubricated at each brake job and they can make a little noise if they run dry. Rain could make them stop squeaking. You can ask your mechanic if he or she lubricated them at the last brake job, and follow their advice if you trust their work and integrity.
Some other part of the brake system such as the master cylinder, the piston in a caliper, a hydraulic hose or a hydraulic line could make a noise from fluid turbulence upon releasing the brake pedal. Very unlikely and not necessarily a sign of trouble. Even the pivot pin at the top of the brake pedal arm could squeak, though none of these would stop when it rains. Since your brakes are working normally and your mechanic sees normal wear and plenty of life left in the pads, in my opinion nothing else needs to be done.
If the brake booster is not the cause and the noise is bothering you, the only way to further diagnose it is with two people: one to step on the brake and another to go around the car with good ears and possibly a stethoscope to find the source.