It's fun to set the world to rights on paper, and SF authors are no slackers when it comes to inventing constitutions.
Some of them are recognizably "close to home" in that they merely tinker with what we already have. Neville Shute's In The Wet (1953) portrays an amendment to one-person-one-vote democracy: everyone still has the vote but the more solid citizens have more than one. I must say I'm attracted to the idea of extra votes being awarded for education, achievement, marriage, etc, with the aim of giving more power to the well-behaved elements of society! Yay! Down with the riff-raff! Sorry. Wasn't me. I never said a word...
Let us proceed to madder and more colourful ideas.
Barrington Bayley, an SF genius if ever there was one, in his Collision with Chronos (1977) describes a free-floating space colony called Retort City, shaped like an hour glass, whose constitution could not have been dreamed up by any other author. The colony's two halves are known officially as the Production Retort and the Leisure Retort. As the terms suggest, one is given to work and to the production of material goods, while the other is given to leisure and aesthetic culture. And neither half envies the other. This despite the fact that they are so segregated, that only new-born babies pass from one to the other! Generations alternate: a father living in the Leisure Retort will shunt his newborn son to the Production Retort and vice versa.
The inhabitants of the Leisure Retort were scarcely aware of the workers who served them, and the workers, in their turn, regarded the participants in the aesthetic leisure culture as idle drones who would probably have been happier doing something useful.
Envy is avoided by the Exchange of Generations.
Each babe was taken from its mother a few hours after birth and transported to the opposite retort, usually to be reared by its paternal grandmother, who previously had surrendered her own child... now the babe's father or mother.
The arrangement was made even more perfect by virtue of the fact that the double exchange could be made simultaneously, even though in real terms a time lag of decades was obviously involved. This was because of the flexible phasing of the two retorts in time. On the same day that a couple parted with their new-born child, they received that child's own offspring... their grandchild.
The protagonist does not share the general satisfaction with this regime. He had been born in the Leisure Retort but his father had hidden him, wanting to keep him, and the deception had lasted for ten years... long enough to make the eventual forced move traumatic. I won't give away any more of the plot. And Retort City is only half the scene of this incredibly rich book.
Another impressive display of political creativity is that achieved by Jack Vance in The Anome (1973). In the subcontinent of Shant on the planet Durdane, the 62 cantons each have their own laws and cultures and have little in common. They are ruled by a nameless unknown, called the Faceless Man because his identity is never revealed. The idea is that no one can suborn a person whom one does not know. Government is cheap, since all the Faceless Man has to do when someone breaks a law is press a button to detonate the offender's torc - a ring containing explosive material that is fitted around every adult citizen of Shant. The Faceless Man, with a couple of assistants, roves the country, enforcing whatever laws each canton has made for itself. Each Faceless Man chooses his own successor; absolute secrecy is preserved.
It makes for a great story though one does wonder how much it could ever work in practice. The same thought will occur to readers of Ursula LeGuin's The Dispossessed (1974), which is a seriously worked-out portrayal of an anarchist society. Could anarchism ever really work? With a change of mindset, maybe it just could! At least, reading the novel I almost thinks so, though with regard to the anarchist aim of utter freedom we are prompted to ask ourselves whether the change of mindset which makes the system possible may acquire an oppressive orthodoxy of its own. The verdict is unsure. The anarchists of Anarres may or may not be heading for political and social failure. We are left to wonder.
Philip K Dick's Solar Lottery (1955) envisages a world state in which the ruler is chosen by chance. It is a whackily gripping novel, and true to its own inner laws, so that its lack of exterior credibility does not matter. It turns out that the so-called chance election of the ruer has been fixed, and not by a really bad guy either; just by someone who was fed up with things as they were. In A E van Vogt's The World of Null-A (1948), a Games Machine appoints the World President and other officials, though here the results are supposed to stem from the tests of ability set by the Machine, rather than from chance. Here, too, it turns out that the process is being tampered with.
The abilities required in the rulers of the Uranian cities described in the Ooranye Project are such that no political system known to man could fit the bill. Uranian rulers - Noads, or "foci" - must have a quality known as lremd, which could be called "steering" or "juggling", the kind of awareness with which anyone who has to multi-task in our world (teachers, mothers, managers) must be familiar. But on Ooranye the need is far greater, and the multi-tasking ability is necessary in its heightened form, lremd. Government on Ooranye tends to be light and cheap, with huge trust being placed in the Noads and especially on the Noad-of-Noads, the Sunnoad. Only exceptional individuals can hope to fill these roles. If a Noad goes bad, no constitutional procedure exists for his/her removal, so something unrecorded has to happen, some violent remedial action by private citizens, with no questions asked. If a Sunnoad goes wrong, the result is more surprising to our mentality. A Sunnoad may be Corrected by someone who dares to use force. If the action is justified, the Corrector is acclaimed as such, while the Sunnoad stays Sunnoad, and the reign is, if anything, enhanced by the successful Correction. But death is the swift penalty for a would-be Corrector who fails.
Lastly may I mention what is surely one of the most loveable of political tales: Double Star by Robert Heinlein (1956), in which an actor is pressurized into impersonating a political leader during a temporary emergency, only to find that he has to take on the role permanently and for real. Interestingly, the Solar System-wide culture described in the novel is a constitutional monarchy in which the imperial throne is occupied by the House of Orange. One of Heinlein's best - absolutely unputdownable.