Why are blue light-emitting diodes important enough to warrant a Nobel prize? I am using a miniature blue light as night bed room light since my childhood. For me blue light-emitting diodes almost look like miniature blue lights. I don't see any fundamental scientific breakthrough in it. So why are blue light-emitting diodes important enough to warrant a Nobel prize?
 A: From the Nobel press release...

This year’s Nobel Laureates are rewarded for having invented a new energy-efficient and environment-friendly light source – the blue light-emitting diode (LED). In the spirit of Alfred Nobel the Prize rewards an invention of greatest benefit to mankind; using blue LEDs, white light can be created in a new way. With the advent of LED lamps we now have more long-lasting and more efficient alternatives to older light sources.

https://www.nobelprize.org/prizes/physics/2014/press-release/
They do not seem to claim that the prize is for the best science but for a use of physics that benefits humanity.
You may not agree with awarding the physics prize for this reason. Perhaps that would be more like a "Nobel Prize in Engineering" or perhaps even the Peace Prize if someone wants to argue that LEDs are good for the environment. However, they do not seem to think that the physics is revolutionary, just that the invention has a major positive impact on humanity.
A: There's nothing that I can see in Alfred Nobel's will that refers to fundamental scientific breakthrough.
What it does say is:
...to be distributed annually as prizes to those who, during the preceding year, have conferred the greatest benefit to humankind. 
The blue LED arguably meets that criterion.
A: In addition to white LED lamps (as given in @Dave's answer)
there is one more technical application.
With the availability of blue LEDs, it became possible
to build LED color screens for TV and computers,
and replace the energy-consuming and bulky
screens based on cathode-ray tubes.
This was not possible by using only the previously existing
red, yellow and green LEDs.
