The band structure and stability of XBi and XBi3 (X  =  B, Al, Ga, and In) single sheets are predicted using first-principles calculations. It is demonstrated that the band gap values of these new classes of two-dimensional (2D) materials depend on both the spin-orbit coupling (SOC) and type of group-III elements in these hetero-sheets. Thus, topological properties can be achieved, allowing for viable applications based on coherent spin transport at room temperature. The spin-orbit effects are proved to be essential to explain the tunability by group-III atoms. A clear effect of including SOC in the calculations is lifting the spin degeneracy of the bands at the Γ point of the Brillouin zone. The nature of the band gaps, direct or indirect, is also tuned by SOC, and by the appropriate X element involved. It is observed that, in the case of XBi single sheets, band inversions naturally occur for GaBi and InBi, which exhibit band gap values around 172 meV. This indicates that these 2D materials are potential candidates for topological insulators. On the contrary, a similar type of band inversion, as obtained for the XBi, was not observed in the XBi3 band structure. In general, the calculations, taking into account SOC, reveal that some of these buckled sheets exhibit sizable gaps, making them suitable for applications in room-temperature spintronic devices.