This is a simple primary battery. Referring to it as a thermocouple or thermopile is misdirection, and there's certainly no need for zero point energy or 'quantum' effects. It's the same thing you could do with alternating disks of copper, zinc, and paper, in a bath of vinegar, but more durable. Some folks out there may remember doing things like that in grade school science class, if you had a fun teacher.
You could do this with any two metals, and just about any electrolyte. In most cases such a battery degrades quickly, as the metal disks react with the electrolyte even when no electrons are flowing. Early batteries of this type were called 'plunge' batteries, because you 'plunged' the battery into the electrolyte when you needed electricity, and lifted it back out when you were done, to keep it from dissolving. This battery has cleverly worked around that by using metals that don't react quickly with the electrolyte.
In this case, if the article is to be believed, it looks like the metals are gold and platinum, and the electrolyte is sulfuric acid. I think this is plausible. Both gold and platinum dissolve only very slowly in acids unless under extreme oxidizing conditions, in which case an oxidizing agent like fuming nitric acid or concentrated hydrogen peroxide oxidizes the metal and then the oxides dissolve in the acid. Those conditions aren't present in this example, so these metals are relatively inert in these conditions, and the reaction is very slow.
It looks like this battery has a fair amount of cells, though I couldn't really find an image clear enough to count them. That makes coming up with a reaction rather difficult. Assuming that they're accurately reporting the 'one volt' output, (exactly one volt?) there are some places where gold and platinum are close enough to each other in the electrochemical series to make a one volt battery with twenty or thirty cells plausible. Without the battery to study, the exact construction and reaction are going to remain a mystery, and it doesn't sound like the museum has any interest in letting this be studied.
It sounds like they're measuring the electrons flowing from the cell with some sort of ballistic galvomometer arranged as a motor. These draw very little current, and even then the article implies that the battery has to recharge between turns of the motor, so it has a mechanism that turns it off after each swing. This is consistant with the idea that this battery operates by very slow reactions.
I see that some folks have already brought up the oxford electric bell. This is the same sort of battery, in that it has electrodes in very unreactive conditions. In that case, the electrodes are probably something quite far apart on the electrochemical series like copper and silver, and they're thin, so that there can be lots and lots of them. Similar examples have hundreds or thousands of layers. The low reactivity conditions are achieved by it being a 'dry pile' in which the electrolyte is at a very very low concentration.
If I recall, the oxford electric bell produces a fairly high voltage at almost unmeasurable current. It's basically a source of 'static electricity' to run a small 'franklin's bell'. You can run a franklin's bell from the current that leaks from the screen of a television; they don't draw much current at all. The low reactivity conditions, combined with the low current draw mean that these things run 'forever'.
No mystery here, and no practical power source either. You'd have trouble making either one of these light an LED.
