The Court of Common Council is the main decision making body of the City of London (the so-called Square Mile), and is attended by the Lord Mayor and Aldermen in all their finery.
They hold a meeting once a month in the medieval Guildhall (sometimes they skip a month, so make sure you check the dates), and whilst the contents of the meeting itself can be quite dry and boring, it is well-worth attending simply to see inside the Great Hall.
The meeting begins with the Aldermen gathering at the top table, before the Lord Mayor processes in from the back dressed in his or her ceremonial robes. After the opening speech the Lord Mayor is largely silent, but the Aldermen will discuss such issues as the City police, car parking, library funding, and anything else that affects the Square Mile. Whilst all of this is going on you can pass the time by looking at the grand statues of Churchill, Nelson, Wellington and Pitt, and the huge golden statues of Mog and Magog glaring down from the balcony. The hall is also hung with lots of ancient flags and shields, representing London's ancient guilds.
Craig has been to the Common Council meeting himself, and definitely recommends combining it with a guided tour of the Guildhall beforehand. The guide will end the tour roughly one hour before the Common Council meeting begins, and will tell you what you need to do in order to get back inside again. Read Craig's review of the Guildhall tour now, and then read his review of the Common Council meeting as well. This will give you a good idea of what you see. Feel free to ask him a question about the Common Council as well.
If you enjoy watching political debates then you're in luck -- because there are plenty of better ones in London. Craig's favourite debate is definitely Prime Minister's Questions, but you need to get yourself an invite for that one, so maybe you'll prefer watching MPs in the House of Commons. Read Craig's review of PMQs, his review of the House of Commons and House of Lords, to get a feeling for the atmosphere inside the chamber. Another good one is Mayor's Question Time at City Hall. Craig has written a review of that one as well.
Craig’s review – “Here's a pub quiz question for you: how many Mayors do we have in London? (Nobody ever gets this right.) One? Two? If you think the answer is two then you're wrong, because the correct answer is thirty four. We have the Lord Mayor of London (covering just the City of London), another one for the City of Westminster, plus the actual Mayor of London who works out of Cit… continued”
Guided tour of the historic Guildhall – 18 Oct, 6 Dec 2018, 10 Jan, 7 Mar, 25 Apr, 23 May, 20 Jun, 18 Jul, 12 Sep, 10 Oct and 5 Dec 2019 – Look around one of London's most historic buildings, with a Great Hall dating all the way back to medieval times.
Listen to MPs in the Houses of Commons – 8 Jan to 20 Dec 2018 – Visitors can enter the Houses of Parliament for free, and watch MPs debating in the House of Lords and House of Commons.
Attend PMQs (Prime Minister’s Questions) – Every Wed – You can watch the Prime Minister get grilled by the House of Commons every Wednesday, at Prime Minister's Questions.
Sadiq Khan, at Mayor’s Question Time – 13 Sep 2018 to 21 Mar 2019 – The London Assembly will be grilling Sadiq Khan at Mayor's Question Time, in their monthly meeting at City Hall.
Listen to a speech at Speakers’ Corner – Every Sun (around noon) – If you visit Speakers' Corner at Sunday lunchtime you can listen to an orator stand up and discuss whatever they like -- all in the name of free speech.
A Tudor Christmas, with Alison Weir and Siobhan Clarke – 19 Nov 2018 – Historical author Alison Weir will be talking with Siobhan Clarke about how the Tudors celebrated Christmas.
Michael Morpurgo, talking about War Horse – 20 Nov 2018 – Michael Morpurgo will be talking about the motivations that led to him writing his West End hit War Horse
You might also be interested in attending these other political debates
Here are some more free events in London if you want a cheap day out
Disclaimer: Event details can change at short notice and you should reconfirm everything before making plans