Many hot Jupiter systems exhibit significant misalignment between the orbital axis of the planet and the spin axis of its host star. While this misalignment could be primordial in nature, a large fraction of hot Jupiters are found in systems with distant stellar companions, and thus could have undergone Lidov-Kozai (LK) oscillations and acquired their misalignment dynamically. Here we present a study of the effect of spin-orbit coupling during LK oscillations, and the resulting spin-orbit misalignment angle distributions. We show that spin-orbit coupling induces complex, often chaotic, behavior in the spin axis of the host star, and that this behavior depends significantly on the mass of the planet and the properties of the host star (mass and spin history). We develop a semi-analytical framework that successfully explains most of the possible stellar spin behaviors. We then present a comprehensive population synthesis of hot Jupiters created via the LK mechanism, and discuss their possible observable signatures.