In this paper, we propose a model to investigate the electron thermionic emission from single-layer graphene (ignoring the effects of the substrate) and to explore its application as the emitter of a thermionic energy converter (TIC). An analytical formula is derived, which is a function of the temperature, work function, and Fermi energy level. The formula is significantly different from the traditional Richardson-Dushman (RD) law for which it is independent of mass to account for the supply function of the electrons in the graphene behaving like massless fermion quasiparticles. By comparing with a recent experiment [K. Jiang et al., Nano Res. 7, 553 (2014)] measuring electron thermionic emission from suspended single-layer graphene, our model predicts that the intrinsic work function of single-layer graphene is about 4.514 eV with a Fermi energy level of 0.083 eV. For a given work function, a scaling of T3 is predicted, which is different from the traditional RD scaling of T2. If the work function of the graphene is lowered to 2.5-3 eV and the Fermi energy level is increased to 0.8-0.9 eV, it is possible to design a graphene-cathode-based TIC operating at around 900 K or lower, as compared with the metal-based cathode TIC (operating at about 1500 K). With a graphene-based cathode (work function=4.514 eV ) at 900 K and a metallic-based anode (work function=2.5 eV ) like LaB6 at 425 K, the efficiency of our proposed TIC is about 45%.