Hexagonal binary intermetallics A(5)B(3) has a unique A(6) octahedra chain structure, providing space for interstitial chemical engineering the physical, mechanical, electrical, and chemical properties without change in the basic structure of crystal. Because of the engineering importance of Zr-Sn alloy, here, we investigate the influence of 24 interstitial alloying elements X (X=B, C, N, O, Al, Si, P, S, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Nb, and Sn) on stability and properties of hexagonal Zr5Sn3 via first-principles calculations. A general trend is that the additional element with small atom size and high electronegativity is favorable as interstitials in Zr5Sn3. The calculated formation enthalpy and the elastic constants suggest that these Zr5Sn3X structures are thermodynamically and mechanically stable. The calculated phonon spectra indicate that Zr5Sn3X structures are dynamically stable except X=V, Cr, Mn, Zn, and Nb. We show that their electronic structures including bonding characters have strong correlation with the stability and mechanical properties. With strong covalent bonds, Zr5Sn3B has the highest Young's modulus, bulk modulus, shear modulus, Debye temperature, and microhardness. The addition of alloying elements decreases the anisotropy except X=O, Sc, Ti, V and Nb. All the additive elements increase the specific heat capacity of Zr5Sn3. Our results could be helpful in designing and improve the performance of Zr-Sn alloy on demand.