My toilet turns on for 5 to 10 seconds every few hours. I assume that there's a leak, that the water level recedes, and when it gets low enough, the float allows the water to come back on. I can't find the leak. I turned off the water at the wall, and drained the tank but there wasn't anything obviously loose or not well seated. I'm stuck!

Put something in the tank that will discolor the water. Food coloring works well. If the color appears in the bowl without flushing the flush valve is leaking.

Have you tried replacing the flush valve and cleaning the surface it seats against? Have you insured the linkage between the flush valve and the handle doesn't have a little tension on it which would lead to it holding the valve open?

Note that the cheapest toilet flush valves at the hardware store or home improvement retailer might be a quality corresponding to their low price. I had one several years ago that was bad when it was brand new. I've found that the extended time between replacements make the more expensive flush valves well worth the added expense.

My toilet turns on for 5 to 10 seconds every few hours. I assume that there's a leak, that the water level recedes, and when it gets low enough, the float allows the water to come back on. I can't find the leak. I turned off the water at the wall, and drained the tank but there wasn't anything obviously loose or not well seated. I'm stuck!

There are several things that can caause this problem The first is a very simple thing to look for, from the main valve in the tank you will have a small 1/4 inch refill tube that runs over to a stand pipe, this stream of water runs in there to refill the bowl after a flush. So here is what you look for, see how far into the 1/4 inch tube goes, it should only be about 1/2 inch and it should always be higher than the level of the water in the tank. So if this is the problem lift the tube up above the water level and secure it someway so that it will stay that way.

Now talking about the level of the water in the tank, after the tank has filled and the water has stopped the level should be about 1/2 to 3/4 of an inch below the refill pipe.

The next thing is the ball that falls into the bottom of the tank to stop the water form running into the tank. Me not knowing what kind you have it is hard for me to tell you exactly what to do but first lift that ball up and feel of the bottom, it should be smooth and you should not get any black on your fingers, if it is rough or makes black, replace it. While you are replacing it feel with you fingers the ring in the bottom of the tank that the ball fits against, make sure it is smooth if not use some steel wool or sand paper an smooth it up, One of these things should solve you problem.
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Check the flapper,the piece on the bottom of the bowl,they do fail and then leak slowly.

The fill valve is the part where the water comes in to the tank,they also fail,but,first check to see if you can adjust the water level by adjusting the fill valve float,just the tiniest adjustment might make your problem disappear.

Sounds like you have problem with you flapper valve, check the flapper and the seat for any distortion; does it have any waved edges? Do your hands get black? Replace it if it does. And last thing check the seat it self. For any nicks or rough spot?

Also you can check if the flapper valve is leaking place some food coloring in the toilet tank, after an hour or so check if dye showed up in the toilet BOWLS then you will know for sure that the flapper valve is defective and needs to be replaced see the image below, good luck.

John