The spectroscopy and thermodynamics of the atmospherically important NaO+(X(3)Sigma(-)) species are calculated using high-level ab initio calculations, up to the RCCSD(T)/aug-cc-pVSZ level. The calculated dissociation energy was Do 0.29 +/- 0.02 eV, which is to the low end of the range of values obtained from previous experimental studies. This lower value for the dissociation energy of NaO+ has implications for experimental values of the dissociation energy of NaO and the adiabatic ionization energy (AIE) of NaO. Further calculations led to the dissociation energy of NaO(X(2)Pi) being calculated as D-0 = 2.71 +/- 0.07 eV, with the AIE for the NaO+(X(3)Sigma(-)) <--NaO(X(2)Pi) ionization being calculated as 7.55 +/- 0.08 eV. The dissociation energy of NaO is to the high end of previous experimental determinations but agrees very well with previous ab initio calculations. The AIE is somewhat higher than the onset measured in a recent photoelectron experiment, and this is discussed. In addition, a potential energy curve and dipole moment function for NaO+(X(3)Sigma(-)) are calculated at the RCCSD(T)/aug-cc-pVQZ level. From the potential energy curve, vibrational separations and rotational energy levels were calculated, and from the calculated rovibrational levels, thermodynamic quantities were derived. The low-energy region of the NaO(X(2)Pi) potential energy curve is also calculated, allowing the lowest energy levels to be obtained. From the vibrational wave functions of the neutral and cationic states, Franck-Condon factors were calculated for the first photoelectron band, which corresponds to the NaO+(X(3)Sigma(-)) <-- NaO(X(2)Pi) ionization. The role of NaO+ in atmospheric chemistry is discussed as well as the implications the present results have on a recently suggested model for the formation of sporadic sodium layers. It is noted that the a(1)Delta state, of NaO+ will be metastable and so may have a role in atmospheric chemistry.