We investigate the thermoelectric properties of the extended Lieb lattice under the influence of intrinsic spin-orbit coupling, on-site Coulomb interaction, and an external magnetic field using the Hubbard model and linear response theory. The unique electronic structure of this system, characterized by two tunable flat bands, provides a fertile platform for controlling charge and heat transport. By systematically analyzing the electronic contributions to thermal conductivity, electrical conductivity, Seebeck coefficient, and the thermoelectric figure of merit (ZTe), we reveal how spin-orbit coupling and strong electron correlations suppress thermal conductivity while enhancing thermoelectric efficiency. Our results show that increasing the on-site Coulomb interaction and magnetic field strength significantly improves ZTe, particularly at low to intermediate thermal energy. The modulation of flat bands via spin-orbit and interaction effects plays a critical role in enhancing the density of states near the Fermi level, thereby optimizing the Seebeck response. These findings highlight the role of flat-band engineered lattices in shaping thermoelectric response and provide theoretical insights into band-structure engineering of correlated low-dimensional materials with tunable transport behavior.
