We determine thresholds p_{c} for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds p_{c} on the coordination number z are tested against various theoretical predictions. The proposed single scalar index ξ=∑_{i}z_{i}r_{i}^{2}/i (depending on the coordination zone number i, the neighborhood coordination number z, and the square distance r^{2} to sites in ith coordination zone from the central site) allows one to differentiate among various neighborhoods and relate p_{c} to ξ. The thresholds roughly follow a power law p_{c}∝ξ^{-γ} with γ≈0.710(19).