Yeah, this is also why an empty hard drive should have less mass than a full one. It's an immeasurable amount, but it should be there. Adding energy to a system should increase it's mass.
To be pedantic, I think you need to say "the act of writing to a hard drive adds mass". There is no reason to expect that writing a 1 or a 0 adds more or less mass; if there is a difference in energy states, it would depend on the encoding scheme.
https://physics.stackexchange.com/questions/31326/is-a-hard-...