We have analyzed the electronic and vibrational properties of two Bi-based cubic Laves phases: RbBi2 and CsBi2. Our results show that the occurrence of superconductivity in these two compounds and an isostructural KBi2 stems almost exclusively from the heavy element pyrochlore lattice. Both the stability of the phases and their electronic and vibrational properties are heavily affected by relativistic effects. The spin-orbit coupling is crucial for the dynamic stability of CsBi2, and to a lesser extent RbBi2. The stability enhancement can be explained by considering an isolated Bi4 cluster and the effects of SOC on its electronic structure. We postulate that crystal structures based on relativistically-stabilized hypervalent networks of heavy elements are interesting candidates for finding new superconductors.