What is this blue thing in a photograph of a bright light? It looks like the bulb of the lamp beside it but how why what

 A: It is the bright light from the bub reflecting off the image sensor in the  camera, then reflecting off the back of one of the lenses and then hitting a different bit of the sensor where it is detected.
Even if the sensor is absorbing 95% of the light hitting it (ideally you want the sensor to absorb 100%) and the lenses reflect only 5% you still have 5% * 5% = 0.25% of the light from the bulb hitting the sensor for the second time - so the blue image is 400x fainter than the white bulb. That's why you only see the effect for bright sources like the light bulb or the sun.
The reason it is blue is likely that both the sensor and the lens have coatings to reduce the amount of reflection. These don't reduce all colors by the same amount, the sensor is also more sensitive at some colors and so might detect fainter blue than red.
A: It's a lens flare of the spiral bulb in the socket.
I assume this is the full uncropped picture. Then the bulb and lens flare position are symmetric with respect to the center of the image (in other words, the optical axis).
Similar lens flares are typically seen in images of the sun, for example in Apollo footage; moon hoax conspiracy theorists get all worked up about them :-)
Here's a picture of such a bulb. If you examine the above picture carefully, you can see the spiral and the diagonal segment.

