We proposed a novel class of honeycomb structures inspired by triply periodic minimal surfaces (TPMS) architecture with tuneable mechanical responses. The design procedure based on the level-set approximation was first introduced for two TPMS-based honeycombs, namely G-Honeycomb and P-Honeycomb. A numerical model was developed and validated by experimental results to evaluate the mechanical properties of G-Honeycomb and P-Honeycomb lattices. The results showed that P-Honeycomb exhibited higher elastic modulus and plateau stress than G-Honeycomb at various relative densities. Meanwhile, both TPMS-based honeycombs showed higher in-plane elastic modulus than conventional square honeycombs, although they had the same nodal connectivity. The idea of designing lattices with tuneable mechanical responses was achieved by introducing a density gradient or creating a G-P hybrid structure with the topology of G-Honeycomb and P-Honeycomb in different regions. The graded structures showed gradual stiffening and progressive collapse under compression, while the G-P hybrid structure can exhibit distinct properties at different compressive strains.