Nothing at all. I know that I am going to be staying basically at St Pauls Cathedral and that is it. My transport for the Red Cross event is going to be paid for so I suspect it will be the bus (or taxi). My best bet would be to get a map and have an explore of the main attractions.
Just as dmoran said, my best bet would probably be to use it for under 30 mins each time.
I reckon Barclays bikes are great for getting around central London.
I can't abide by the Boris moniker though ;-)
I live not far from St Pauls within the hire bike area and find them really convenient. Being in the congestion zone the traffic is kind of ok.
If you're looking to explore and get a feel for the layout of the city, well the obvious thing is to follow the river to Tower Bridge in one direction and to the Houses of Parliament in the other.
You might want to spot the Bank of England, Monument, Tower of London, the London Assembly, Borough Market, The Globe, Tate Britain en route. Certainly with a Barclays bike it's a bit more acceptable to pootle on the pedestrianised parts. It's perhaps worthwhile having a map marking the docking stations.
North of Embankment, there's Trafalgar Square, Whitehall, the Royal Parks, Buckingham palace, Hyde Park, the Royal Albert Hall etc. and if your aim is to get a feel for the layout of London and perhaps cherry pick a few specific places to visit, I would definitely recommend them.