Interesting answer from the New York Times to this question a few years ago (specifically about ebook readers),
“In principle, the answer is yes,” said John D. Kubiatowicz, a professor of computer science at the University of California, Berkeley.
“However,” he said, “the amount is very small, on the order of an atogram,” or 10–18 grams. “This amount is effectively unmeasurable,” he went on, since even the most sensitive scales have a resolution of only 10–9 grams. Further, it is only about one hundred-millionth as much as the estimated fluctuation from charging and discharging the device’s battery. A Kindle, for example, uses flash memory, composed of special transistors, one per stored bit, which use trapped electrons to distinguish between a digital 1 and a 0.
“Although the total number of electrons in the memory does not change as the stored data changes,” Dr. Kubiatowicz said, the trapped ones have a higher energy than the untrapped ones. A conservative estimate of the difference would be 10–15 joules per bit.
As the equation E=mc2 makes clear, this energy is equivalent to mass and will have weight. Assuming that all these bits in an empty four-gigabyte Kindle are in a lower energy state and that half have a higher energy in a full Kindle, this translates to an energy difference of 1.7 times 10–5 joules, Dr. Kubiatowicz calculated. Plugging this into Einstein’s equation yields his rough estimate of 10–18 grams.