As Reach4 says, it's probably worth checking the water pressure. If that were the problem, what would happen is that it would get high enough to push past the seal of the fill valve (which lets water into the tank), and then the water level would rise above the normal level and down the "overflow riser", which is the tube that keeps the tank from spilling over the top if the fill valve stops shutting off. Arguably, when you flush the toilet by pushing the button, the water in the tank drops (because now the toilet is actually flushing), causing the fill valve to open all the way as it normally does (instead of straining against the excessive pressure) and then shuts off when the tank refills. It sounds from your story that once you push the button, the thing is fine until the next morning, so that's a plausible theory.
The other possibility (and the usual problem with "ghost flushing") is that the FLUSH valve seal wears out, water drains from the tank bit by bit, and eventually the fill valve opens to replace the water that was dripping out.
I don't know what parts the plumber replaced. On that toilet, a $3 seal is usually all it takes to fix the fill valve, if it was wearing out. Most likely, he replaced the rubber seal on the flush valve unit and nothing more, because normally nothing more is needed.
Doesn't sound like he stopped to try and actually diagnose the thing. And I hope he didn't charge you to replace ALL the parts, because that's just lazy and unnecessarily expensive.
If you have an inventory of what he replaced, from his bill, that might help us. If you want to take a picture of the inside of the tank and post it, that might help too. It always worries me when Just The Next Plumber works on what I assume is an Aquia dual-flush, because it does have some special parts, and just dropping any old part in there probably isn't going to end well.
(PS If it does turn out that your house water pressure is spiking late at night, the usual culprit is the expansion tank that sits near the water heater. Your water heater heats the water, which makes it expand. Since your pipes don't expand, the expanding water needs somewhere to go. If you didn't have an expansion tank (in a closed system, which I expect you have), the water pressure will spike because the water is getting bigger and has nowhere to go. So you have an expansion tank that is effectively a pressurized bladder, which takes in the expanding water when it causes the pressure to exceed a certain level. After a certain period of years, the expansion tank needs to be serviced or replaced. If it stops working, then it is as if you didn't have one. Once the water exceeds a certain pressure, it is going to start pushing through the weakest valve in the system. Usually, this ends up being an outside water spigot which will start to drip or stream a little water, or often an older toilet fill valve, as the seal on the valve is starting to wear so it can't hold back the the high pressure. So if you get that little gauge and discover that the water pressure is peaking (over say 65 PSI) late at night, then I would look first at the expansion tank to make sure it is operating properly. One reason stuff like this happens is that at the hour you are talking about, you will most likely have stopped using sinks, showers, the dishwasher, etc., and maybe your dishwasher was running or someone was taking a shower as the last use of water for the night, so the water heater had to come on and heat the water significantly, but now it is doing so without any faucets likely to be opened during the time it is doing the heating. Arguably, once you get up and flush the toilet, it relieves the excess pressure and the water heater not having to come back on, you're set for the night.) It's a theory.
