I use full synthetic 10W30 in mine, and change it per the manual. Since generators use relatively little oil, I use the best I can due to the fact that most generators use splash lubrication, so breakdown of oil harms them much more than a system that is pressurized.
Honda markets an approved oil for their gen sets and small engines(Home Depot). I just use it and change every 100 hrs. Yea, I know, you can buy other oil cheaper. For as much as is required why bother with buying a gallon of super premium last all never fail green, yellow, blue, or silver container of whatever oil. Besides, you get just enough to fill the gen if you buy their branded oil in their small containers. Less storage space, easy to handle, small spout.
2010 Ford F-250 Super Duty 4x4
2010 Montana Mountaineer 324RLQ
As long as the oil is the correct weight and is an API grade oil it will work as well as any other oil. Synthetic helps if you plan on operating your generator in sub-freezing temperatures as it can be a lower weight but that is the only advantage.