I was given a 1 day visa to the UK because I had an overnight layover in London as a result of an airline mistake. I'm supposed to fly out today, but I'm running late, and I'm wondering what happens if I miss the last flight out again? Will they give me another one night visa, or will they make me sleep in the airport this time?
Exit checks are being phased in right now. This answer assumes you will encounter one. If you do not encounter one, all bets are off until you need another UK visa. You have until midnight on the expiry date of your visa to leave. If you are still in the UK after that, you will be an overstayer and liable for removal.
It is unlikely that they will simply issue another 1 day visa, the decision they make depends highly upon your nationality and your behaviour. You would need to show some credible evidence of how you spent your day in the UK, and they will run a check for crimes in your locality. You'll most likely be served an IS151A removal order and escorted to detention to await a flight out. Alternatively... If they are busy and your flight is leaving shortly, they will temporarily detain you and escort you to the gate. They will give your passport to a flight attendant. In either case you won't be back for a while.
As commentary has suggested, it's better to approach them before your time runs out. It adds credence to your intentions. Calling the contact centre doesn't count towards improving your credibility; you need to present yourself with your travel document. If they decide to get your biometrics on file, don't refuse or matters will become much worse.
You'd be best to ask the immigration people directly, for an authoritative answer.
UK Visas and Immigration contact centre
Telephone: 0300 123 2241
Textphone: 0800 389 8289
Monday to Thursday, 9am to 4.45pm
Friday, 9am to 4.30pm