Diesel should really be called heavy duty oils. They are not as limited as petrol only oils in terms of limits to additive packages to protect catalytic converters. If you are not going to use synthetic oils, something like at least a 15W-40 or better (like Brock suggests) is good. The diesel oils are probably the most cost effective to use, particularly if you want to change the oil earlier than the Honda recommended interval. A 20W-xx oil I found was too thick when cold for Canberra in winter, but in warmer areas/summer would be good.
I have used the 10W-40 mineral oils for quite a while, and they don't tend to be as good for gear changes as anything 15W-xx or more for 10,000kms plus oil changes.
I have never had oils tested (may as well just change it!) so my comments are really based on the transmission and how it feels when it gets hot - the noisier it gets, the more worn out the oil is.