Which would be better, a MOSFET transistor, or a relay?
Hmm, can you describe what you mean when you say electric strike? I'd have to further understand what you were creating, but I would also go with cheaper and simpler. Both are switches, but relays are electromechanical, while MOSFETs are solid state, which means that mechanical aspect is gonna cost you so cheaper is almost always the transistor. But for instance, if you wanted to electrically isolate your circuit, or were considering the voltage drop across your switch, a relay might better suit your needs.

Always look at your datasheet to determine the current that can flow through your device. If you're trying to use the relay or transistor as a method to turn on or off an actuator that moves a door lock make sure that the current ratings of both the relay or MOSFET could handle it. Other than that, because this project is not too intense, experiment and break things, see what works for you and is easiest. But I would recommend a cheap MOSFET that can handle a drain current of whatever your actuating servo requires.