Lets face it, if you want to fly to London you don't fly to Birmingham and you certainly won't be paying £25+ for a train to London as well no matter how fast it is. At no point have I wanted to fly to Los Angeles and flown to San Francisco and then jumped on a train, the same goes for any other major city in the world. Nice has a good train link with Paris but you wouldn't fly there and then jump on the train if you wanted to go to Paris.
The only clear choice is to build a new runway, it is the cheapest choice and really the only one that makes logical sense. Anyone saying otherwise is missing the fact that it is the largest airport in Europe operating at 99% capacity, it is just a total no brainer.
The Americans usually have it right on things like this, they just build 4+ runways and get it sorted. We are the only ones who can hold up major infrastructure builds because of the effect on birds or a load of stupid idiots who decided it was a good idea to move next to an airport and complain about noise.