As long as its broke in, any oil you like will work. Synthetic oil is not the best to use on new engines not driven yet, its supposedly to slick and it doesn't let the rings seat properly. If the engine has been driven, rings seated, bearings seated, etc.... then use any oil you like. If you are still in the break-in period, then use a mineral oil with some break-in additive added, oil is again your choice. If this engine has a flat tappet camshaft, then use any oil of your choice with additive added (each oil change), till you are tired of buying it. The cam profile and valve spring rate will determine how much additive you will need each oil change. The milder the cam and springs. the less you need. I would use at least half a bottle each oil service with mild cam, full bottle with high horsepower engine.
If this engine has a roller camshaft and lifters, use any oil you like with out additives.
The oil choice depends on what you like, weight doesn't need to be more them 10w40. I have been using 5w30 or 10w30 lately in new engines, the lighter weight splashes around better keeping the cams well lubed, at least in my mind it does.
Joe