toilet
Sometimes my toilet starts running for a few seconds at a time, like it's just finishing filling up the tank.
Usually this happens when the bathtub is being drained, or when I turn on the facets in the sink, but I think it also happens sometimes when no water is being used anywhere else.
As far as I can tell the flapper in the tank is working fine, and I don't see any leaks anywhere.
I'm guessing the valve assembly inside the toilet is finding a resonance when it's just about to close. You can replace it, and it's not too difficult of a DIY job. They sell just the valves:
But if you're going to take it apart to replace the valve, I tend to buy the full kit and replace all the guts.
Note that when my toilets started to leak, it was because of high pressure in the water lines. So it wouldn't hurt to pickup a pressure gauge that screws onto a hose faucet to make sure you don't have a bigger issue. After I fixed one, another would start leaking next as the pressure got a little higher.
Check for any debris around the flapper and its hinges. It is the flapper moving back to the original closed position that puts the handle back in the correct position, not the handle that moves the flapper (when closing, obviously when you flush the handle opens the flapper). You have the right idea with adjusting the chain to ensure there is always slack.
There are usually a couple positions for the chain to attach on the handle itself - try a different position.
If all else fails, try replacing the flapper.
Best Answer
The typical toilet tank will have a flapper which seals the water in the tank. When you flush the toilet, the flapper is lifted and water is allowed to move from the tank into the bowl. Once a set amount of water has exited the tank, the flapper falls back down and re-seals the tank.
The most common issue with toilets randomly flushing is that the flapper has become brittle or sediment has formed on the flapper/tank which prevents the flapper making a complete seal. This will allow water to slowly drip from the tank into the bowl. If enough water is permitted to leak out of the tank, the filling mechanism will be triggered and the tank will refill. If enough water enters the toilet bowl, the self-siphonage will be triggered and the bowl will drain into the sewage system.
Here's a picture of a typical assembly: