In this study we simulate structural, electronic and phonon properties of MgAl2O4 spinel containing single oxygen interstitial (Oi) per crystalline L4 and L8 supercells, e.g., its dumbbell configuration formed with one of the nearest regular oxygen atoms of the lattice (Oi-Oreg). Due to the splitting of the Wyckoff positions in supercell models of perfect crystal, five possible Oi positions with different site symmetry have been identified and studied (C1, Cs, C3v D2d and Td). First principles hybrid HSE06 DFT functional calculations on perfect and defective spinel structures have been accompanied by geometry optimization. The calculated properties of spinel crystal (lattice constants, bulk modulus, band gap as well as frequencies of infraredand Raman-active vibrational modes) qualitatively well agree with the corresponding experimental data. The formation energy of Oi is found to be minimal for the interstitial site of lowest symmetry (C1). The results obtained are important, in particular, for understanding the radiation and chemical stability as well as other key properties of MgAl2O4 spinel-type oxide crystals.