Many stars produce stellar winds that are made up of particles that emanate from stellar surfaces and undergo rapid acceleration, flowing out and shaping the environments of planets and other celestial bodies orbiting the star. Since most stars spin, an interesting consequence of stellar-wind outflow is that angular momentum (or rotational energy) is lost from rotating stars and transferred to the outflowing wind, which causes stars to 'spin down' significantly over their lifetimes. This effect can have consequences for the space environment of planets over long time-scales, and potentially even influence the habitability of planets to host life. This study examines the effect of fluctuations and turbulence in stellar winds on the angular momentum carried by them, focusing on the wind produced by our very own star, the Sun. We develop a mathematical theory that describes this effect, and use a computer simulation of the solar wind together with near-Sun data from NASA's Parker Solar Probe mission to find that the contribution of turbulence to angular momentum tends to be small (but not negligible) and negative in sign. This could imply that turbulence reduces the rotational energy lost by the Sun to its wind over long timescales.