This is probably just a message hooked into the odometer to remind you to change the oil every n thousand miles.
There is really no substitute for checking your oil the old fashioned way, in my opinion. Computers can't account for all driving conditions, what kind of oil you may have put in there, the age of the engine, and other factors.
On older engines where you get more blow-by gases getting gas into the crankcase, you can smell the gas in the oil. I always smell the dipstick when checking my oil. The oil can thin out and lose its lubricity before turning very dark.
I use a white paper towel and examine the oil when I wipe the dipstick. If it looks really dark, is too thin, or smells like gas, it's time to change it.
Synthetic oil is definitely better than regular. It has consistent molecule size, better temperature stability and performance at temperature extremes, and lasts longer. 3000 miles is a good estimate for older cars. Newer ones can go longer, maybe 5-10 but it really depends. Again, there's no substitute for checking it yourself.
Mobil 1 full synthetic and zmax additive have kept my civic running smoothly > 210k miles on the same engine.