As poisoned chalices go, this one is surely deadlier than most. You are a famous architect and win a competition to design for parliament. Sounds good. Only it’s not a megamillion pound building they want — oh no. They want you to stage an exhibition that explains parliamentary democracy to the people. So your task is to sex up a decidedly unpromising dossier. Oh, and it’s got to be in Westminster Hall, where famous corpses get laid out in state. Blimey.

The architect in question is Nigel Coates, last in the public eye for his Body Zone at the Millennium Dome. Coates is also known as the architect of the famously short-lived National Centre for Popular Music in Sheffield — although the building has survived and is now being converted into a student union for Sheffield Hallam University.

Coates, a likeable and highly theatrical architect, has a knack of getting along in the corridors of power. Before the Dome, it was he who masterminded a canonic New Labour moment, the 1998 Powerhouse UK exhibition in a temporary building in Horse Guards Parade. This turned out to be the high point of the phenomenon briefly described as Cool Britannia, back when Blairism was still enjoying its long honeymoon.

House to Home, as the new exhibition is called, is a far cry from all that. It is not a New Labour project, but a product of the cross-party Hansard Society, a serious-minded group that exists to promote parliamentary democracy. How to bring parliament to the people and tackle apathy? The exhibition sets out to do two things: to explain the way the system works now and to suggest ways it might work in the future.

Coates being Coates, this is not your conventional exhibition with display panels. It is laid out, slightly askew, down the centre of the venerable hall, as an appropriately two-chamber affair. It’s a low-budget exercise — and that shows — but Coates juggles his relatively few props with aplomb. In the first section, rows of benches and wooden chairs represent the oppositional debating chamber of today, while in the second (reached through a pink plastic arch), a congenial circle of Coates-designed seating represents a new spirit of inclusive democracy.

In both chambers, you get big videos. In the first, it’s edited highlights from the House — Sir Geoffrey Howe resigning with deadly effect, the former Labour leader John Smith making mincemeat of the Tory government, Maggie Thatcher’s hysterical last Question Time, Blair justifying war, Howard attacking Blair — all explained with a subtitled commentary for the politically tone deaf. In the second chamber, the idea of a public vote on a smoking ban in public places is explored, with contributions from the likes of David Hockney (“I smoke for my health. My mental health.”) You can push buttons to vote, see the results a few minutes later and the resulting mocked-up TV coverage. You can leave your comments on the whole idea of parliament on Post-It notes in booths around the edge. All these will be read and digested for a report on this bit of public consultation. Finally, you can even write and post a question to the prime minister; a few of these will apparently receive a reply. Faced with this possibility, I wrote feebly: “Do you agree that it is important to admit when you have made a mistake?” That’ll cause him sleepless nights.

The show does try a bit too hard. Dress it up as you will, this is always going to be a worthy-but-dull subject. In their eagerness not to look patrician and official, the organisers have gone in for some of that fake street-style lettering, aimed presumably at kids. Nobody’s fooled. Not in the august surroundings of Westminster Hall, right next to Mr Speaker’s state coach.

Verdict? Fun to drop in on if you’re visiting parliament anyway, though I wouldn’t arrange a whole trip to London just for this. The interactive website offers a bit more. For instance, you can feed in your personal details and preferences and it will select the MP who most resembles you. Well, I did that and it gave me the grande dame of Hackney North, Diane Abbott. Should I worry? Or should she?

House to Home is at Westminster Hall until August 20