The hardest question I've answered is "How are probabilities supposed to work in a multiverse where everything that can happen does happen somewhere?" It's hard to say which tricks helped the most because when I started I didn't have a list of tips and tricks, so I don't know how much it would have helped to try to apply them consciously. But here's what worked in retrospect, in rough order of importance:

Go meta. In this case the meta question was much easier than the object-level question, because I could get the answer from history. Probability theory was created by gamblers, and later formally justified using decision theory, so I knew I should take a decision theory approach to the question.

Don't stop at the first good answer. Here is the first good answer that I might have stopped at. (The website was created by Hal Finney some years ago.)

marks's Solve many hard problems at once. Yep, I was also trying to answer "Does quantum immortality/suicide make sense?" and "How are probabilities supposed to work when mind copying is possible?"

Be ready to recognize a good answer when you see it. Apparently several lesswrongers have discovered the same answer independently, but I was the only one who thought it was a big deal and wrote it up. Others shrank from its counter-intuitiveness, or just didn't realize its significance. I also discussed the idea on my own mailing list, where it failed to make much of a splash.

Explore multiple approaches simultaneously. and Trust your intuitions, but don't waste too much time arguing for them. The main approaches were "first-person" and "third-person", and my approach is mostly third-person, but I also spent a lot of time thinking about the first-person approach. (The first-person approach is more concerned about expectations of subjective experiences.) I think there were too many arguments about which is the right approach, when the time could have been better spent actually exploring them.

Sleep on it. Pretty hard to say how much this helped, but I did often go to sleep thinking about the problem.