Funny, I wondered about the same thing a year or two ago and ran across this:
Honda Engines FAQ
I might use synthetic in Honda lawn mowers and such, but for something as critical (and finicky) as a generator I'll stick with what it was designed to use - and maintain it (including oil changes) as recommended in the manual. Of course, that's just me.
This says to go by the standards that is in the manual. If you check the standards in the manual and the synthetic oil meets those standards you are good to go. The motor was designed to an oil standard and not a specific type of oil. Most oils you find on the shelf at Walmart or any parts store will meet the standards of most engines.