We develop an effective Gauss-Bonnet extension of loop quantum cosmology, by introducing holonomy corrections in modified F(G) theories of gravity. Within the context of our formalism, we provide a perturbative expansion in the critical density, a parameter characteristic of loop quantum gravity theories, and we result in having leading order corrections to the classical F(G) theories of gravity. After extensively discussing the formalism, we present a reconstruction method that makes it possible to find the loop quantum cosmology corrected F(G) theory that can realize various cosmological scenarios. We exemplify our theoretical constructions by using bouncing cosmologies, and we investigate which loop quantum cosmology corrected Gauss-Bonnet modified gravities can successfully realize such cosmologies.