What is an airliner's "green dot speed", and how does it vary with weight, and why?
Green Dot (GD) speed is a term used by Airbus. It is named after the symbol on the speed tape. It refers to the speed that results in the best climb gradient in case of an engine failure, but it is close the speed resulting in best L/D ratio with all engines operating.

Airbus defines it as follows:


  Definition
  
  GD speed is the engine-out operating speed in clean configuration. In
  other words, it corresponds to the
  speed that allows the highest climb
  gradient with one engine inoperative
  in clean configuration.
  In all cases (all engines operative),
  the GD speed gives an estimate of
  the speed for best lift-to-drag ratio. It
  is also the final take-off speed and it
  represents the operational speed of
  the clean configuration and the recommended speed in holding in clean
  configuration.
  It is represented by a green dot on the
  PFD speed scale and displayed only
  when the slats / flaps control lever is in
  the ‘0’ (CLEAN) position and landing
  gears are not compressed
  
  
  
  How is GD determined?
  
  GD speed is computed by the Autoflight systems and is based on the
  aircraft weight. The GD formula has
  been set up so that the resulting airspeed provides the best lift-to-drag
  ratio for a given altitude, air temperature and aircraft weight, in clean configuration with one engine out.
  In some phases of flight, GD is computed to minimize drag and thus, the
  fuel consumption (for example during
  the HOLD phase).


(Airbus Safety First Magazine)

The GD speed is increasing linearly with aircraft weight and slightly increases with altitude above FL200:


(Airbus A320 QRH - 4.01 In Flight Performance - Speeds)

Here is a plot of the GD speed as a function of weight for different altitudes: