Mobil makes the stock honda oil and its not synthetic if you are seeing it for $5 or less. Up here in Canada, they have honda synthetic/normal oil right on the shelf side-by-side at the honda shop. So synthetic oil will not harm your engine...as long as it has the WET CLUTCH CERTIFICATION.
Its the wet clutch certification that is the absolute key. If it doesn't have it I wouldn't put it in. Some opinions about energy conserving may/may not be true, but for me, oil is the absolute life blood of these air-cooled engines, and isn't worth the risk. Oil does everything; lubricate moving parts, cool the engine and allow for proper clutch operation. We spend $100-1000's on tires, pipes, filters, axles etc and yet quibble about a low dollar item that if missing the correct additives causes more damage than everything else, and no proper running engine=no fun.
Look at it this way, you spend more on gas than you do for oil during its oil change interval...to me, no contest.
Certified oil, synthetic/normal oil both have their advantages. Normal oil is inexpensive and can facilitate more frequent changes & good for intermediate climates. Synthetic oil has much better high and low temperature protection. Up here in Canada, its synthetic all year around. Easy cranking in sub-zero temps & if you are mudding, don't have alot of airflow around the engine, or are in hot climate, I wouldn't use anything but synthetic. Look at Redline, Amsoil, its found everywhere-even on-line, its wet clutch certified, and guaranteed, if you talk to any racer they have it in their engine.
Put the oil designated and wet clutch certified for the engine whether normal/synthetic flavors, and no worries..... My .02