Don't rent. Just buy.
Mobile phones are dirt cheap in the UK.
Search and you will see lots of previous on this.
There are stores called Carphone Warehouse all over the place. You can buy their cheapest ones for 4 pounds and then buy 15 pounds of minutes.
They're absolutely right. Just buy one. It's way cheaper and it's yours to do as you wish. You can always sell it on ebay when you get home or donate it when you leave if you don't care to keep it.
There is a Carphone Warehouse on Earl's Court Road, not far from the junction with Cromwell Road. There is another on Kensington High Street, just across the road from M&S and near Ryman. Speaking of Ryman, they often sell inexpensive Sim cards, so pop in to see what's on offer.
There is also a branch of Argos - www.argos.co.uk - on the High Street, oppo side from the Underground station and just west of the junction with Argyll Street if I remember correctly. The number 49 bus towards Hammersmith conveniently calls right in front of Gloucester Road station on its way to Kensington High Street and in front of the Stanhope Arms on its way back.
Swapping out SIM cards is a good idea, IF your particular cell phone is set up to allow that. Not all are, mine isn't.
For your information, here are the prices of phones in the Argos stores, which are very common.
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