I've been using Mobil 0W30 or Castrol Syntec 0W30 in my "good" car for years. At first the "0W" scared me a little, but after doing some research the light bulb lit up. The zero-w viscosity rating refers to the viscosity at mid-winter cold cranking temperatures. In fact, the viscosity curve of these oils is such that they are actually thicker at temperatures above the operating temperatures of conventional 5W30 or 10W30 oils, i.e., they don't thin out as much as conventional oils under extreme high temperature conditions.
So, yes, use a synthetic in your mower if that makes you feel better. However, these motors are not stressed as high as racing engines, so whatever the manufacturer recommends will work just fine. However, if you're on the habit of running your engine out of oil, synthetic will give you an extra margin of safety.