Information to be stored or manipulated requires energy, any energy in the storage system contributes to its mass as does raising the temperature of the storage system. This effect is negligible in any calculations.
Different configurations of charge or magnetic or molecular arrangements in storage mediums represent different energies so in effect you can change the mass of a storage device depending on what is stored there. The amount of energy represented is negligible and may not be directly proportional to the amount of data stored. So information is not measurable in mass and the mass changes in typical devices is likely not measurable either. Also depending on how information is actually stored you may be decreasing the energy (and thus mass) when going from "blank" to "programmed".
To complicate matters a very small change in a typical modern storage mechanism could represent a very large change in the quantity of data stored.
There have been a few estimates that the mass of all of the moving electrons that make up the internet at any given time is about 50 grams.
There are two answers here. Assume the internet contains about 2000 petabytes of data - that is 2 million terabytes, or 16 billion gigabits.
2 million terabytes would fit onto about 500,000 pounds of solid state devices - assuming about 4 ounces per terabyte. That's data you could actually access. But it's not the answer you want.
If you transmitted the entire internet, how much energy (and thus mass) would be used to encode that data? Assume 16 billion gigabits of data, a 1 watt laser diode, and a 1 gigabit per second fiber line that is perfectly efficient, unbreakable and as long as needed. It would take 16 million kilowatt/hours to transmit the Internet. That's 5.76x10^13 Joules (1.6x10^7 kilowatt/hours) of energy to transmit the Internet through that cable. That works out to about 64 grams of mass for the energy encoding the Internet.
It would take you 500 years to transmit the Internet. The fiber that would hold the Internet from start to finish would weigh in at 1.55x10^20 pounds (a hundred billion billion pounds). It would wrap around the Earth 1.23x10^14 times.