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Hexagonal germanium in the lonsdaleite structure has a direct band gap, but it is not an efficient light emitter due to the vanishing oscillator strength of electronic transitions at the fundamental gap. Transitions involving the second lowest conduction band are instead at least three orders of magnitude stronger. The inversion of the two lowest conduction bands would therefore make hexagonal germanium ideal for optoelectronic applications. In this work, we investigate the possibility to achieve this band inversion by applying strain. To this end we perform ab initio calculations of the electronic band structure and optical properties of strained hexagonal germanium, using density-functional theory with the modified Becke-Johnson exchange-correlation functional and including spin-orbit interaction. We consider hydrostatic pressure, uniaxial strain along the hexagonal c axis, as well as biaxial strain in planes perpendicular to and containing the hexagonal c axis to simulate the effect of a substrate. We find that the conduction-band inversion, and therefore the transition from a pseudodirect to a direct band gap, is attainable for moderate tensile uniaxial strain parallel to the lonsdaleite c axis.