Some want to construct a third runway, but that would likely require the demolition of entire villages, as Heathrow is in a dense area outside London.
There is a more outlandish idea that takes advantage of unused space: Build a new airport, and make it float on the estuary of the Thames River. One group pushing for this option, the Thames Estuary Research and Development Company (Testrad) has released new plans for its "innovative and spectacular" London Britannia Airport.
It certainly is spectacular. Here's the view from the air:
And a panoramic rendering (on the next page):