Can I use stratified random assignment in a factorial experiment?

Yes, in a limited way. First, make sure you understand the answer to the FAQ “How do I randomly assign participants in a factorial experiment?”, which describes how to create lists of random numbers for random assignment purposes. You can conduct stratified assignment easily by maintaining more than one list of random numbers. For example, if you want to stratify by gender, you can maintain one list for males and one list for females. Note that stratification has the effect of ensuring that if the subject sample is, say, 40% male, each experimental condition will contain approximately 40% males.