Technically flash memory storage would weigh more as a 1 (or maybe as a 0, depending), since it works by storing electrons. Apparently it stores something like 500 electrons [1]. My phone has 6 GB of storage, so if they are all ones that gives an extra weight of 2.50e-13 g, which is 0.0001 nanograms. Probably not measurable, though, especially since the large mass of the phone relative to the change in mass eliminates a lot of sensitive measurement techniques.
I'm not sure if RAM would have extra weight, as I know less about how it works.
I know you're being funny, but researchers in a lab got an encryption key by listening on the spinning discs if I remember the paper correctly. Was front-page few years ago.
Like that one time I set all RAM in my phone to 1s and couldn’t lift it anymore. /s