You have a fairly easy problem. When a toilet fill valve tries to shut off filling the the tank after a flush, the diaphragm inside the top of the fill valve closes the opening of the fill valve (called the seat). The seat is a small metal or plastic orifice that is about 3/8 inches in diameter. The water travels up through the center of the fill valve an through the seat. As the float lifts, the diaphragm is pushed down and forms against the seat and the water shuts off. Since the diaphragm is made out of rubber or neoprene it is soft and pliable, however when it is in the water for a period of time, it dries out and is no longer pliable. It becomes like a super ball..hard and dense. Once it tries to form around the seat, it can't quite shut the water off, so it bounces in micro bounces. These bounces set up a harmonic wave inside the copper or steel water piping. The piping amplifies it just like a tuba or a trumpet.
What's happening at night, is your flapper is allowing a small amount of water to leave the tank and turning on the fill valve to refill it. So really you have two problems. The reason it comes on when you use any other fixture, is because you have lowered the pressure in the house which sends a wave through the water system and the fill valve starts bouncing again.
The simple solution for this is to replace your fill valve. A relatively easy project.
1. Turn off the water to the toilet.
2. Remove the supply tube.
3. Empty the toilet tank.
4. Remove the lock nut holding the fill valve inside the tank.
5. Buy a ffluid-master400A anti-siphon ball **** (about $12).
6. Buy a new Korky flapper (non water saver).
7. Buy a new flexible supply tube Fluid master.
8. Install the Fluid master according to the directions 1" above the overflow tube.
9. Install the flapper.
10. Install the supply tube.
11. Turn on the toilet.
I think this will take care of your problem.
I always suggest replacing the fill valve rather than the diaphragm, because it is a much easier project. In addition, I suggest replacing the flapper at the same time, because they are the same age (typically) and yours seems to be half the problem. Make sure you take your flapper along with you so you can match up the right one. They are all different. You need to get the flapper rated for the gallons in your tank or your toilet won't flush right. Finally, if you make sure you replace the supply tube with a flexible one, you will not have a leak under the toilet.
Always read all directions that come with the flapper, fill valve and supply tube. There are specific tightening instructions and measurements inside the boxes.
Raymond VinZant Plumbing Prof.