We investigate the spin-dependent thermoelectric effect of graphene flakes with magnetic edges in the ballistic regime. Employing static, respectively, dynamic mean-field theory we first show that magnetism appears at the zigzag edges for a window of Coulomb interactions that increases significantly with increasing flake size... We then use the Landauer formalism in the framework of the non-equilibrium Green's function method to calculate the spin and charge currents in magnetic hexagonal graphene flakes by varying the temperature of the junction for different flake sizes. While in non-magnetic gated graphene the temperature gradient drives a charge current, we observe a significant spin current for hexagonal graphene flakes with magnetic zigzag edges. Specifically, we show that in the "meta" configuration of a hexagonal flake subject to weak Coulomb interactions, a pure spin current can be driven just by a temperature gradient in a temperature range that is promising for device applications. Bigger flakes are found to yield a bigger window of Coulomb interactions where such spin currents are induced by the magnetic zigzag edges, and larger values of the current. (read more)