Just because work is done doesn't mean the energy is stored. It could just result dissipate as heat.

As a counter-example, consider etching data in some form into another material, say stone or metal. You do work to remove the material you etch away, but you are removing material, so the final mass is actually less than you started.

That said, I believe most digital storage uses a high energy and low energy state to store 0 and 1, and in that case the high energy state will have (very, very slightly) more mass than the low energy state. But even then, having all bits in the high energy state would be the "heaviest", but would effectively have no data.

Work being done to it will show up as heat, which indeed is subject to e=mc^2. But when it cools there's no residual mass.

The data doesn't actually have weight because they aren't going to store a 1 or a 0, but rather do something like store 01 vs 10.