Quasifree photoproduction of eta mesons off nucleons bound in the deuteron has been measured with the CBELSA/TAPS detector for incident photon energies up to 2.5 GeV at the Bonn ELSA accelerator. The eta mesons have been detected in coincidence with recoil protons and recoil neutrons, which allows a detailed comparison of the quasifree n(gamma,eta)n and p(gamma,eta)p reactions. The excitation function for eta production off the neutron shows a pronounced bumplike structure at W=1.68 GeV (E{gamma} approximately 1 GeV), which is absent for the proton.