Hydrogenation reactions at gigapascal pressures can yield hydrogen-rich materials with properties relating to superconductivity, ion conductivity, and hydrogen storage. Here, we investigated the ternary Na-Si-H system by computational structure prediction and in situ synchrotron diffraction studies of reaction mixtures NaH-Si-H-2 at 5-10 GPa. Structure prediction indicated the existence of various hypervalent hydridosilicate phases with compositions NamSiH(4+m) (m = 1-3) at comparatively low pressures, 0-20 GPa. These ternary Na-Si-H phases share, as a common structural feature, octahedral SiH62- complexes which are condensed into chains for m = 1 and occur as isolated species for m = 2, 3. In situ studies demonstrated the formation of the double salt Na-3[SiH6]H (Na3SiH7, m = 3) containing both octahedral SiH62- moieties and hydridic H-. Upon formation at elevated temperatures (>500 degrees C), Na3SiH7 attains a tetragonal structure (P4/mbm, Z = 2) which, during cooling, transforms to an orthorhombic polymorph (Pbam, Z = 4). Upon decompression, Pbam-Na3SiH7 was retained to approx. 4.5 GPa, below which a further transition into a yet unknown polymorph occurred. Na3SiH7 is a new representative of yet elusive hydridosilicate compounds. Its double salt nature and polymorphism are strongly reminiscent of fluorosilicates and germanates.