Black-hole (BH) binary mergers driven by gravitational perturbations of tertiary companions constitute an important class of dynamical formation channels for compact binaries detected by LIGO/VIRGO. Recent works have examined numerically the combined orbital and spin dynamics of BH binaries that undergo large von Zeipel-Lidov-Kozai (ZLK) eccentricity oscillations induced by a highly inclined companion and merge via gravitational wave radiation. However, the extreme eccentricity variations make such systems difficult to characterize analytically. We present an analytical formalism for understanding the spin dynamics of binary BHs undergoing ZLK-induced mergers. We show that, under certain conditions, the eccentricity oscillations of the binary can be averaged over to determine the long-term behavior of the BH spin in a smooth way. In particular, we demonstrate that the final spin-orbit misalignment angle θsl is often related to the binary’s primordial spin orientation through an approximate adiabatic invariant. Our theory explains the “90∘ attractor” (as found in recent numerical studies) for the evolution of θsl when the initial BH spin is aligned with the orbital axis and the octupole ZLK effects are negligible — such a “90∘ attractor” would lead to a small binary effective spin parameter χeff ∼ 0 even for large intrinsic BH spins. We calculate the deviation from adiabaticity in closed form as a function of the initial conditions. We also place accurate constraints on when this adiabatic invariant breaks down due to resonant spin-orbit interactions. We consider both stellar-mass and supermassive BH tertiary companions, and provide simple prescriptions for determining analytically the final spin-orbit misalignment angles of the merging BH binaries.