Does light have mass/weight? If I plug in my memory stick/USB into my laptop and load it up with documents etc. will its mass/weight increase?
 A: You confuse between four different concepts:


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*First, between mass and weight.


Yes, mass is not weight. They are two different concepts: mass is a physical property of matter which determines the body's resistance to being accelerated by a force, for example: It requires more force to make a car move than a bicycle, it's because the car has more mass than the bicycle.
Weight is the gravitational force that pulls on mass. On the Moon, your weight is reduced compared to your weight on earth because gravity is weaker there, but your mass is still the same.


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*Second, between light and electricity.


Light is an electromagnetic wave, example: the light coming from the sun.
Electricity is not due to electromagnetic waves, but because of a negatively charged subatomic particle, namely the electron. Everything around you is made up of different kinds of atoms which are themselves composed of a dense nuclei and of electrons. Electricity is the flow of those electrons, from an atom to another. So when you charge your smartphone via USB, the electrons flow from the computer, to the wire, to the battery of the smartphone. 
Light doesn't have mass but electrons do, their mass is: $9.11×10^{−31}\text{kg}$. In fact, when you load a USB with documents or when you charge a phone, its mass increase by a tiny tiny amount, around $10^{-18}\text g$. It's because of changes in the level of energy stored of the electrons on your phone, their physical number stays the same but they possess more energy, hence more mass, $E=mc^2$ as Einstein cleverly putt. 
To continue in our discussion about electromagnetic waves: Light has no mass, but it do possess momentum, hence it has weight because gravity is caused by momentum and not mass.
A: Re. 2nd question:-
"If I plug in my memory stick/USB into my laptop and load it up with documents etc. will its mass/weight increase?"
Its mass will not necessarily increase or decrease.  It depends on the prior state of the memory.  If there are the same average numbers of binary ones and zeros before and after the weight would be approximately the same.  This is due to how the ones and zeros are stored on the memory stick.  Placed electrons create charges on the chip gates to store information, but if you are just rearranging them there is no extra weight.
Ref. Wiki:- Floating Gate Transistor

"Because the floating gate (FG) is electrically isolated by its
  insulating layer, any electrons placed on it are trapped there and,
  under normal conditions, will not discharge for many years."

