Method and content in selecting a social science dissertation project General question: When working in an area of social science where method matters just as much as content (i.e., theory, question), how does one choose which should be the starting point for a dissertation? In other words, should one (option 1) choose a question that is of great theoretical interest/relevance and then try to find compelling methods or (option 2) choose a question that can be answered using cutting-edge methods and then frame it in the literature as well as possible? 

This question matters because, although a project will need some of both in the end, the starting point of a dissertation will determine which one of the two goals above is more feasible. I recognize that it takes years of hard work to achieve just one of those two goals. Also, the answer cannot simply be "both," since that just means waiting to find a methodologically great project that has theoretical relevance. As I can see it, that's just a very successful execution of option 2.

Personal specifics: I'm a PhD student in political science in the US exploring potential dissertation topics. (I applied with a research topic but, because I'm in the US, I'm free to change it in the first few years of the program.) I work in a niche of political science that places a lot of value on "causal identification" using a certain set of statistical strategies. This means that most books and articles celebrated by my advisors use some statistical strategy to estimate the effect of an intervention on an outcome. This usually requires that the researcher find some intervention that was randomly assigned (e.g., by nature, or a government, or an NGO, etc.). 

Despite the general emphasis on method, my advisors say that I should pick a question/puzzle/topic first. Also, I have spoken with the authors of some of the aforementioned books, some of which grew from dissertations. The authors mostly state that they picked the question first and later luckily happened across their identification strategies. Part of me wonders if they are just saying that because it is the way that one "should" start a research project. (Unfortunately, I'm not friends with these authors so I cannot have a totally frank conversation them.) 

In general, it seems more intellectually honest to select a question or topic first and then find good methods for studying it. But it generally seems more feasible to select a question that I expect I can answer with a cutting-edge technique first, and figure out how to frame the question later. How does one manage this tension at the beginning of a dissertation project or (I suppose) any long-term research project?
If I understand your question, it sounds like you would prefer to identify a topic or research question of interest, but feel pressure to ensure that question fits with a specific statistical technique. 

Your dissertation project is about showing that you are capable of conducting research, but it is still under the guidance of your adviser. If they want you to use a certain technique, then that's part of the training process. Once you're an independent researcher you may decide to approach the issue differently, but I'm not sure there's anything inherently dishonest in this process.

One option is to identify the key components needed to use that technique (x number of variables, access to a large population database, etc.), pick a broad topic or research question you want to answer, and then look for datasets that will allow you to address some aspect of that question. This way, you can identify the topic and theoretical justification for why it is worth studying, while leaving room to narrow your eventual research question based on the available data which matches this technique. 

Keep in mind that as a student you are still learning these techniques and how to integrate them with your research questions. Also, the authors you mentioned  likely have years of experience in their field. It's possible they've reached a point where they think of/tailor their research questions to fit within a specific statistical framework, even if they aren't aware of it. (Imagine if the statistical technique you used for every study was regression; eventually, you would probably begin thinking of research questions in terms of regression). Perhaps by working with these methods and then identifying the specific question, you'll gain the flexibility to "reverse-engineer" the process when you have more experience.