A spin ensemble in thermal equilibrium continuously undergoes random fluctuations analogous to those observed in Brownian motion, a process known as spin noise. Here we investigate the dynamics of a system comprising a spin-1 paramagnetic center and a hyperfine-coupled spin-1/2 nucleus in the vicinity of a level crossing. We theoretically show that nuclear spins polarize efficiently under the combined action of thermal fluctuations and a closed-loop feedback protocol. The latter articulates periodic observations of the electronic magnetization and radio-frequency pulses connecting hybrid states with opposite nuclear spin alignment. Since nuclear polarization emerges from electronic spin fluctuations, not spin order, this microwave-free technique generically benefits from warmer, not colder, operation temperatures. Further, because the spin dynamics at play near a level crossing is rather insensitive to the absolute value of the magnetic field, our work promises opportunities for high-field dynamic nuclear polarization, difficult to attain through present methods.