Three new Ag(I) metal-organic frameworks (Ag-MOFs) containing bisbenzimidazole-derived ligand, namely, {[Ag2(L)(TP)](H2O)}n (1), {[Ag2(L)(HBTC))(H2O)](CH3CH2OH)(H2O)}n (2) and {[Ag4(L)2(IP)2](CH3CN)3(H2O)16}n (3) (L = bis(1-(pyridin-4-ylmethyl)-benzimidazol-2-yl methyl) ether, TP = Terephthalate, BTC = Benzenetricarboxylate, IP = Isophthalate), have been synthesized by volatilization method and systematic adjustment of solvent and aromatic polyacid co-ligand. Single crystal structural analysis showed that although three Ag-MOFs have a three-dimensional (3D) network backbone structure, their topological structures are different. Electrocatalytic hydrogen evolution reaction (HER) of carbon paste composite electrodes (Ag-MOF-1∼3@CPE) prepared by mixing graphite powder with Ag-MOFs were investigated in 0.5 M H2SO4 electrolyte. The HER measurements show that the overpotential η10293K of Ag-MOF-1∼3@CPE are -634, -533 and -615 mV compared with sCPCE (blank electrode, -966 mV), and the Tafel slope of sCPCE and Ag-MOF-1∼3@CPE were 276, 251, 123 and 220 mV dec−1, respectively. The results prove that Ag-MOF-1∼3@CPE could effectively catalyze and accelerate the HER behavior, with electrocatalytic activity order being Ag-MOF-2@CPE > Ag-MOF-3@CPE > Ag-MOF-1@CPE > sCPE. The highest HER activity of Ag-MOF-2@CPE can be attributed to the fact that Ag-MOF-2 contains coordination water molecule, while the higher activity of Ag-MOF-3@CPE than Ag-MOF-1@CPE is due to steric hindrance. Furthermore, the recognition performance of Ag-MOF-1∼3@CPE for H2O2 was further studied by chronoamperometry in 0.2 M phosphate buffer solution (PBS, pH = 6). The three sensors can detect H2O2 in a linear range from 0.5 μM to 4 mM with sensitivities of 103, 34.5 and 138 μA·mM−1·cm−2, and also revealed long-term stability and good selectivity. Our study provides a new approach for designing efficient, non-precious metal electrochemical catalysts.