We demonstrate how redox control of intra-molecular quantum interference in phase-coherent molecular wires can be used to enhance the thermopower (Seebeck coefficient) S and thermoelectric figure of merit ZT of single molecules attached to nanogap electrodes. Using first principles theory, we study the thermoelectric properties of a family of nine molecules, which consist of dithiol-terminated oligo (phenylene-ethynylenes) (OPEs) containing various central units. Uniquely, one molecule of this family possesses a conjugated acene-based central backbone attached via triple bonds to terminal sulfur atoms bound to gold electrodes and incorporates a fully conjugated hydroquinonecentral unit. We demonstrate that both S and the electronic contribution Z(el)T to the figure of merit ZT can be dramatically enhanced by oxidizing the hydroquinone to yield a second molecule, which possesses a cross-conjugated anthraquinone central unit. This enhancement originates from the conversion of the pi-conjugation in the former to cross-conjugation in the latter, which promotes the appearance of a sharp anti-resonance at the Fermi energy. Comparison with thermoelectric properties of the remaining seven conjugated molecules demonstrates that such large values of S and ZelT are unprecedented. We also evaluate the phonon contribution to the thermal conductance, which allows us to compute the full figure of merit ZT = Z(el)T/(1 + kappa(p)/kappa(el)), where kappa(p) is the phonon contribution to the thermal conductance and kappa(el) is the electronic contribution. For unstructured gold electrodes, kappa(p)/kappa(el) >> 1 and therefore strategies to reduce kappa(p) are needed to realize the highest possible figure of merit.