It that true, though? Does an electronic storage device with full storage have more electrons in it? I don't know enough about how modern memory works to say for sure.

Electrons are used as the information storage mechanism in flash memory, but the data is whitened (50 % one bits on average independent of the data input) before being stored. Hence, "the number of electrons" would not vary between an SSD storing only zeroes, and one only storing ones.

But in any case, you would be measuring the mass of electrons and not the information they represent. This theory suggests that a storage device would have additional mass dependent on information content, so things like whitening / encryption would not matter.

However, here is a question that immediately occurs: If I have a message with "informational mass" x, and encrypt it, does the encrypted message has the same informational mass, or is the mass of the encrypted message the same as a random message of the same length?

Since flash is based on a charge pump, doesn't it just move the electrons from the substrate or control gate to the floating gate? Or am I missing something here?

In that case you do not have to account for the electron mass at all.

Electron rest mass is about 1E-31 Kg which is orders of magnitude more than the proposed estimated bit mass of 1E-38 Kg.

That's not really the point. The point is, they would be measuring the weight of the electrons, in other words, the implementation-specific medium used to store the information. Not the information itself.

I can see this discussion go very far into metaphysical. :)

Well, what is information? Is it an abstract thing, or does it only exist in terms of the thing which is representing it?

I don't have a clue, I just think it's a relevant question at this point.