Economic: Looking at the common factors, it's about high vs low barriers to entry. The idea of non-entry is thus an obvious place to look. The first thought is a "gift economy". Riffing off the idea of the Bayesian Conspiracy, we get furtive students exchanging notes in dark alleys: a world where economy is forbidden. The exact nature of what resources are available to each Bayesian Initiate are left as an exercise for our FAI Overlords, but should presumably be sufficient to avoid students being incapable of building the requisite experimental apparatuses for their latest project.
Sexual: Again, focusing on the common ground, the difference is "readily available" vs "tightly restricted". Thinking about fun theory, the idea of sex as some sort of reward seems obvious - presumably one related to self-understanding and building relationships. Everyone is situated such that they have a genuine route to satisfaction, whatever their desires may be. I'd suspect desires are also tweaked somewhat downwards - it's more a driving force like curiousity, not an urgent biological imperative. It's rewarding to pursue, not just to receive (but, like curiousity, it works best when you actually achieve the goal). Unsure what durations look like, but things are well-enough tweaked that there aren't violent break-ups when one partner loses interest. Probably sexuality is something that ebbs and flows over time, letting people focus on deep, intimate sharing with people at times, and letting them get lost in other challenges at other times.
Government: Centralization vs Democratization of power; the tyranny of the minority vs the majority. Both are seeking to enforce morality on those who desire to do "wrong". Both are seeking a universal system of moral authority. Certainly, a large chunk of disagreement can be resolved between the combination of being led to understand our own extrapolated volition, and the elimination of scarcity. When there are no factual disagreements, and no suffering from the status of others, then I think it would take a fairly abnormal human to still desire non-consensual violence. This still feels more like a fixed utopia than any sort of weirdtopia though.
Technological: Being lazy, we invoke the Bayesian Conspiracy: you start with next to nothing, and can benefit from those technologies you're able to implement yourself. Given a few thousand years, you ought to have computers back! Please consult economics for the question of resource scarcity; we probably don't want to hand anyone who asks for it a chunk of U-238, and it wouldn't be terribly polite to start the students off with nothing more than an unlimited supply of hydrogen (although if they were the omnipotent administrator of a simulated universe, and could thus build stars, then planets, then life, I'd find the resulting projects utterly fascinating!)
Cognitive: For this one, let's assume the Bayesian Conspiracy doesn't surpress anything developed 21st century or earlier. Computers offer prospective Initiates obvious ways to exponentially amplify their understanding of science, and at a certain Level of the Conspiracy, you have to start showing pretty impressive exponential growth. Since you have to understand a problem before you can build a tool to solve it for you, you now have the ability to automate large chunks of the "boring" parts of science. There's still the joy of discovering more-optimized algorithms, although this will only be relevant for programs with fairly long run times, or for the inherit beauty that comes with elegance. Any unfriendly AI, grey goo, etc. is stopped by the FAI overlords and thus the world is kept safe. This is considered a failing mark, and you are summarily stripped of all the knowledge you've learned from the Conspiracy and forced to start over as a Zeroth Level adept once again. Developing a Friendly AI is required to reach Level 10, at which point you can start doing "raids" (group-based adventures, for those unfamiliar with WOW :)). The FAI Overlords, for whatever reason, can only reproduce via this method, and place a strong value on the diversity this adds to their ranks. Guided AI research is, for whatever reason, unsatisfying to them - it has to be someone who went from pre-science to FAI all on their own. Fortunately, the FAI Overlords aren't omniscient, so some cheating does occur (see Economics). Higher level adepts are usually much better at cheating, which helps prepare them for doing raids.
Economic: Looking at the common factors, it's about high vs low barriers to entry. The idea of non-entry is thus an obvious place to look. The first thought is a "gift economy". Riffing off the idea of the Bayesian Conspiracy, we get furtive students exchanging notes in dark alleys: a world where economy is forbidden.
Ever read The Disposessed by Ursula Le Guin? About half of it is set in a society in which "asking for something in exchange" is considered highly immoral.
Utopia and Dystopia have something in common: they both confirm the moral sensibilities you started with. Whether the world is a libertarian utopia of the non-initiation of violence and everyone free to start their own business, or a hellish dystopia of government regulation and intrusion—you might like to find yourself in the first, and hate to find yourself in the second; but either way you nod and say, "Guess I was right all along."
So as an exercise in creativity, try writing them down side by side: Utopia, Dystopia, and Weirdtopia. The zig, the zag and the zog.
I'll start off with a worked example for public understanding of science:
Disclaimer 1: Not every sensibility we have is necessarily wrong. Originality is a goal of literature, not science; sometimes it's better to be right than to be new. But there are also such things as cached thoughts. At least in my own case, it turned out that trying to invent a world that went outside my pre-existing sensibilities, did me a world of good.
Disclaimer 2: This method is not universal: Not all interesting ideas fit this mold, and not all ideas that fit this mold are good ones. Still, it seems like an interesting technique.
If you're trying to write science fiction (where originality is a legitimate goal), then you can write down anything nonobvious for Weirdtopia, and you're done.
If you're trying to do Fun Theory, you have to come up with a Weirdtopia that's at least arguably-better than Utopia. This is harder but also directs you to more interesting regions of the answer space.
If you can make all your answers coherent with each other, you'll have quite a story setting on your hands. (Hope you know how to handle characterization, dialogue, description, conflict, and all that other stuff.)
Here's some partially completed challenges, where I wrote down a Utopia and a Dystopia (according to the moral sensibilities I started with before I did this exercise), but inventing a (better) Weirdtopia is left to the reader.
Economic...
Sexual...
Governmental...
Technological...
Cognitive...