Well, this may not quite fit the personal criteria for a wierdtopia, since this is quite close to what I would consider a utopia, but it stands fairly far apart from the other scenarios presented so far, so I figured I might as well post it:
Tech Everyone is forcibly uploaded, the surface of the earth is scanned in super-duper-hi-fi precision and then used for computronium to house the newly uploaded minds. An overseer AI is created that sends out a sphere of near-light speed probes to convert the rest of the stuff in our future light cone into computronium.
Cog Everyone stays more or less at the level they are, since they are all supposed to pull themselves higher mentally by their own will, and almost everyone is too busy screwing around for the foreseeable future.
Gov't Doesn't exist, since everyone has their little (or rather huge) chunk of cyberspace where they can summon any simulacra of beings that they want. The overseer AI has discovered that there is no verifiable difference between conscious and unconscious algorithms (but the scanned in humans are almost definitely conscious) so it is satisfied with providing everyone with a way to create human simulacra that are probably not conscious and allowing the probably conscious humans to do whatever it is they want to them (simulating earth histories is a popular hobby).
People are allowed to visit other people's created worlds, and some even do it quite a lot. Violence can only happen if the victim allows themselves to be hurt, and those conflicts that can't be resolved without emotional damage (eg love triangle) are handled as they are now, except without the possibility of suicide (well, you can do it in theory, but only if the overseer analyzes you and sees that you would never recover from your emotional loss, and that's never happened so far). At most, the hurt party researches neuropsychology for a while before invariably getting distracted.
Econ Similarly doesn't exist except for what people want to make up in their fake worlds. The overseer oversees (duh) the actual RL acquisition of material for more computronium, and distributes it to the measly amount of humans that want to use even a small fraction of it.
Sex One of the biggest distractions. This field got researched to a fairly decent level, so people get to have many of their craziest fantasies fulfilled. To uploaded people, the difference between another uploaded human and a simulacrum is academic, so most create elaborate worlds for their satisfaction. Many also have real loving relationships, but things like "public" perceptions of sex don't really play a role in anyone's life.
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...