Judging by recent questions, I realise this forum really lacks a thread with updated information on this topic. It's a good idea to start one and hopefully people will add to it. I am not that up to date but I know of the following offers:
Most UK prepaid SIMs are completely free if you can get them posted to you. Otherwise they can be bought very cheaply in most shops, usually only at the cost of the credit on them. Some are even being given away at airports and important tourist areas.
Some UK prepaid plans will give you a free bundle of data just for topping up with £10 (or in some cases less). Offers vary all the time, and sometimes you get other freebies (like SMS) for topping up with prepaid credit.
o2 give you a fee SIM and a £10 topup entitles you to free SMS and internet (I believe up to 500MB) to spend within the first month.
O2 Free SIM.
T-Mobile also will give you a free SIM but instead of data give you free SMS for topping up. With them, data on the handset is charged at £1 per day. I believe there are still some older deals that give you free data for topping up.
T-Mobile free SIM card
Three will give you a free SIM, and used to give you some data for that but not anymore. See:
Three free SIM
The biggest advantage with three is that they give you free Skype access which does not count towards your data allowance. Data is charged as follows:
GiffGaff will give you a free SIM too, and until the end of this month will get you completely free data. Thereafter, it will be charged at a cap of 50p/day for up 30MB. It will be charged at 20p/MB up to the cap, and after 30 MB/day.
To get a SIM see:
GiffGaff Free SIM
Hopefully someone else can add in some info about other providers. I seem to have run out of time.
A very useful reference for all things to do with mobile data (including coverage) in the UK is this:
Mobile Broadband ? reghardware