I am starting an internship in London starting October and was wondering what the options for a short-term mobile plan in the UK are.
I use an iPhone so a data plan would be nice but not necessary.
The "Text & Web" tariff from O2 Pay as you Go might be a good option for you. http://www.o2.co.uk/tariffs/payandgo
Topping up £10 in a month gives you 300 free UK texts and 500MB UK web and Wi-Fi and you get to keep the £10 as credit on your phone to use for calls and additional texts.
The text and data allowance runs out at the end of the month but the credit remains.
However, if you mainly plan to use the phone for calls, a rolling (1 month) contract from a network like Three might be a better option for you: (see the £10 and £12 1 month contracts) http://store.three.co.uk/SIM_Only/iPhone_Micro_SIM
Basically to get a reasonable deal you need to have an unlocked phone, so make sure your iPhone is unlocked before you come.
And then you need to do your homework and check all the main providers to see what they have on offer at the time you are ready to buy it - it does change quite often. The main operators are: Vodafone, O2, Three, T-Mobile, Virgin Mobile, there are some smaller ones as well.
You are probably either interested in a pay as you go option or pay monthly (rolling 30 day) contract.
It's not a well publicised network, but a cheap one for PAYG (with unlimited data) is Giffgaff.
It uses the O2 network, and keeps the costs low by not having any phone customer support, it's all forum and online ticket based. £10 gets you 250 minutes, unlimited texts and unlimited data (but unlike the O2 offer the £10 is spent on this offer, it's not still available to you as credit). You get £5 credit on a new SIM card, and they can only be received by post.
The link above is a direct link to their website, but one of the offers that they do is to reward members for referring new users. This following is my affiliate link (assuming I'm allowed to post it here).