We present a statistical reconstruction framework for space-based extreme ultraviolet (EUV) ionospheric tomography. The EUV technique offers a means to invert the nighttime F region electron density on global scales from a single spaceborne spectrograph, using prominent optically thin emissions produced by radiative recombination of O+. Since the EUV technique does not rely on ground receivers to make the measurements, the observations do not suffer from limitations on the viewing angles. The EUV tomography is an ill-conditioned inverse problem in the sense that its solution is sensitive to perturbations of the measured data. With large condition numbers of a typical projection matrix, simple least squares inversion techniques yield unacceptable results in the presence of noise. This reflects the fact that more degrees of freedom are being sought than are supported by the noisy data. To overcome this limitation, we cast the tomographic inverse problem in a stochastic framework and incorporate a statistical prior model. In doing so we also obtain measures of estimation uncertainty for the solutions. Through simulations, we demonstrate the applicability of these techniques in the context of a space mission designed for EUV ionospheric tomography, namely, the Tomographic Experiment Using Radiative Recombinative Ionospheric EUV and Radio Sources (TERRIERS). The simulations show promising results for EUV tomography as a viable ionospheric remote sensing technique.