I would pull it out every month, run it 20 minutes or so, and put it back in the storeroom. It has, until recently, run smoothly. Yes, I put in only enough fresh gas to run it a bit, and ran it nearly dry each time. Then I would close off the fuel valve and run it out.

I took it out to use for Field Day and it ran rough. I had to pull the choke out a bit to get it to run smooth and not "hunt", but it did run all day.

I took it to a nearby Honda authorized service dealer, that sells and services Honda generators and outboard motors, and a number of other generators and outboards (Mercury, Suzuki, Yamaha, others).

The service dealer is a trustworthy outfit and were well familiar with the problem.  They told me, and I was told the same by others, the problem I and others are having is due to ethanol in the fuel. Apparently the place where I get gas has recently changed to gas containing ethanol and that was the cause of the rough running problem. They also said the engine would run fine on pure gasoline.

The upshot is to use a fuel stabilizer, and if possible, use only real gas, not gas with ethanol.  Steve K9ZW (a pilot) said to use Avgas as it will have no ethanol if I can't find a gas station selling straight gas.

Also, I need to run it more often and longer when I do run it.

The final thing... they increased the midrange jet by .001". The jet was not clogged, it was just too small for the fuel obtained here. The ethanol leaning the mixture could be compensated for with modern computer controlled engines but not with this generator. The ethanol was just enough to lean it too far. Richening the midrange fixed it. They still recommend using straight gas if I can get it.

So the generator is working smoothly now, and I'll give it a better test run tomorrow if it doesn't rain again.

I've also picked up some Stabil (fuel stabilizer) and tomorrow will head out to the airport for some Avgas.
 
Further research, we don't have a choice... "E10" (10% Ethanol gasoline) is being forced on us.  Pretty soon all gas stations will have nothing but.
Hope this helps you if your generator starts to run in the "rough".

Update: