This idea has been alluded to before. Not sure about the island concept, but the whole idea of building it at that kind of place was mooted.
There are environmental sensitivities in the area which I think are going to make building an airport there impossible (estuaries easily lend themselves to that). Not to mention creating artificial islands is going to draw similar ire anyway (where exactly are you going to get the landfill to make the islands...) As Mal alluded, bird strikes were also an issue (and you just can't kill all the bird population there...)
The other London airports aren't exactly full (except LCY is already getting full too), but most of the capacity problems will be in the London airspace, not slots on the ground (i.e. not how many planes can land or take-off, but purely just how many planes you can have hovering around London at any given time). That is starting to peak out, too. You could kill off the curfew at the moment, but curfew times don't usually attract a lot of pax (the new demand will probably be filled with LCCs who will fly at that time for the sake of lower fees, hence driving demand in that way), and in any case if we look at SYD we can attest to how successful politically it would be to eliminate a curfew...
London's a basketcase, in short. Encouraging the growth of regional hubs, especially to avoid London airspace, with high-speed connections to London is probably a better idea, though most people of course won't like it (why do I have to fly into bl**dy East Midlands then take a train to London when I could've just flown straight to London). London could probably also stiffen the slot ballots and force higher bidding for slots so that'll kill off the small fish who are just in London because they want to fly to London, freeing up slots for those airlines who will be prepared to put money where their mouth is. But that's probably anti-competitive, killer for LCCs, will force up airfares and won't make many people (passengers) very happy.