It's more complicated than that. A single light switch stores exactly 1 bit of information. Its either on or off and both positions convey the exact same amount of information.
Furthermore, information is related to interpretation. A string of bits may be an uncompressed text file, a set of numbers, a compressed image, etc. But it's always the same amount of bits. How can my interpretation of that set of bits affect its mass?
There's quite a bit of difference between how we usually think about information and how information theory treats it. There it's all about entropy.
As I said to the above poster I don't know much about this, so hope this isn't an off-base question.
The meaning of a certain sequence of bits depends on interpretation, of course. But nobody is talking about meaning here, only entropy. For our purpose it wouldn't matter if there were no sentient beings in the universe to attribute any meaning to anything.