If you're adding energy into the system then it gets heavier under mass-energy equivalence. In the same way raising an object increases the mass very slightly due to the energy stored as potential energy.

For instance, flash storage would increase in mass. You're storing electrons which as matter has its own mass, but you also have to put those electrons into a higher energy state by doing work, which also generates its own mass.

A hard drive is changing the orientation of magnetic grains, which by itself should not increase the energy levels.

In an organic system like the brain, the amount of mass would change to a far greater extent just by normal physiological processes.

Short version: adding energy to something adds mass, but only if this energy is stored.