At the tagged photon facility PHOENICS at the Bonn accelerator ELSA a measurement of the target asymmetry of the reaction gamma p --> p eta from threshold to 1150 MeV has been performed. Simultaneously the reaction gamma p --> p pi(0) has been measured in the first resonance region. Results are presented for both reactions. The target asymmetry data are suited to put considerable constraints on the model parameters used for the theoretical description of meson photoproduction.