Birnessite compounds are stable across a wide range of compositions that produces a remarkable diversity in their physical, electrochemical, and functional properties. These are hydrated analogs of the magnetically frustrated, mixed-valent manganese oxide structures, with general formula, NaxMnO2. Here we demonstrate that the direct hydration of layered rock-salt type α-NaMnO2, with the geometrically frustrated triangular lattice topology, yields the birnessite type oxide, Na0.36MnO2⋅0.2H2O, transforming its magnetic properties. This compound has a much-expanded interlayer spacing compared to its parent α-NaMnO2 compound. We show that while the parent α-NaMnO2 possesses a Néel temperature of 45 K as a result of broken symmetry in the Mn3+ sublattice, the hydrated derivative undergoes collective spin freezing at 29 K within the Mn3+/Mn4+ sublattice. Scaling-law analysis of the frequency dispersion of the ac susceptibility, as well as the temperature-dependent, low-field dc magnetization confirm a cooperative spin-glass state of strongly interacting spins. This is supported by complementary spectroscopic analysis [high-angle annular dark-field scanning transmission electron miscroscopy (TEM), energy-dispersive x-ray spectroscopy, and electron energy-loss spectroscopy] as well as by a structural investigation (high-resolution TEM, x-ray, and neutron powder diffraction) that yield insights into the chemical and atomic structure modifications. We conclude that the spin-glass state in birnessite is driven by the spin frustration imposed by the underlying triangular lattice topology that is further enhanced by the in-plane bond-disorder generated by the mixed-valent character of manganese in the layers.