We investigate different approaches towards a nonvolatile switching of the remanent magnetization in single-crystalline ferromagnets at room temperature via elastic strain using ferromagnetic thin film/piezoelectric actuator hybrids. The piezoelectric actuator induces a voltage-controllable strain along different crystalline directions of the ferromagnetic thin film, resulting in modifications of its magnetization by converse magnetoelastic effects. We quantify the magnetization changes in the hybrids via ferromagnetic resonance spectroscopy and superconducting quantum interference device magnetometry. These measurements demonstrate a significant strain-induced change of the magnetization, limited by an inefficient strain transfer and domain formation in the particular system studied. To overcome these obstacles, we address practicable engineering concepts and use a model to demonstrate that a strain-controlled, nonvolatile magnetization switching should be possible in appropriately engineered ferromagnetic/piezoelectric actuator hybrids.