Hyperconjugation involving other bonds
Your description of hyperconjugation is a very minimalistic one. More generally, hyperconjugation should be seen as the interaction of occupied bonding MOs with unoccupied ones — these unoccupied ones can be both antibonding MOs or empty (cationic) p atomic orbitals.

In principle, $\ce{Si-H}$ or $\ce{Si-Si}$ bonds would also be able to donate electron density in a hyperconjugative matter. In pracitce, due to silicon’s low electronegativity and greater size these orbitals simply do not overlap as well and their energy levels are such that less stabilisation may be obtainable.

Carbon-silicon bonds are of course great at providing hyperconjugation since silicon — much like hydrogen — has a lower electronegativity than carbon, meaning the occupied orbitals have a greater contribution on carbon.