The hydrogen ground-state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach, we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work, we consider the regularization of the noise in the hydrogen ground-state problem. Several renormalization schemes are considered. Some are well-behaved, whereas in others the high frequency renormalization induces pathologies at low frequencies. In no situation did we find a way to escape from the previously signaled self-ionization.