Family Weirdtopia: Individual autonomy pairs with a lower-energy, more localized existence built on the framework of dense, diverse urban population centers. The importance of kinship in human social systems diminishes gradually. Projected far enough out: Lineal descent is tracked solely through the direct relationship between mother and child; you do not share a family name with your grandparents or grandchildren. One's romantic and sexual relationships are not typically expected to outlast their natural lifespan; it's always nice when a couple finds themselves together after years or decades, but to pursue that as a goal would be seen as a strange, somewhat immature obsession or possibly an uncommon fetish. Despite this, social units and parent-child families are not isolated; the old saw "it takes a village to raise a child" is taken for granted, and your neighbors, friends and community members are likely to take a direct interest in your children's upbringing. Committed "mentor" and "caretaker" relationships are provided by means of various social institutions to those whose community ties haven't naturally offered up people willing to specifically dedicate to those roles.
Economic Weirdtopia: Bioregional commonwealths become the dominant world-system. This isn't a utopia; one's lifestyle, political and social opportunities become increasingly sharply-defined by where one lives, as do elements of culture and economic opportunity. Trade flows naturally across ecological and economic boundaries, and the existence of air and sea travel mean long-distance trade and travel remain viable, but globalization is a thing of the past, as is standardization. Life is very good in your little corner of the world as long as you fit there comfortably, but if you don't it's a long and difficult process to just relocate. It's not a simple matter of gathering your things and moving, after all -- your destination community needs to have room and someone to sponsor or support you during the transition to a very different way of life. International finance as we know it is dead in the water, but many forms of poverty and lack are a thing of the past. Cultural insularity is checked only to the degree that the citizens of a given Commonwealth engage with folks outside their milieu online.
Cognitive Weirdtopia: Standard cultural practice is to treat most forms of pedagogy or personal development like forms of budo (http://en.wikipedia.org/wiki/Bud%C5%8D#Civilian_vs._military). Early childhood education more closely resembles a freeform series of monastic-type practices guided by one's mentors; the emphasis is on the disciplined practice and refinement of whatever idea is being taught; understanding is the pupil's job. The less-pernicious aspects of self-help culture are seen as basic common sense; most people "know themselves" very well and by adulthood, are very skilled in several major disciplines (though "disciplines" spans the entire spectrum of human endeavor). Rationality is not inherently prized; it is more natural to most people to understand their failure modes and plan life around them rather than strive to change -- which is usually seen as a risky endeavour, less laudable when successful for the sheer inadvisability of teaching the average person to do so for it's own sake.
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...