On the disk an individual bit weighs nothing, it's just a change in magnetic polarity; see TheTXI's answer for a more elaborate explanation of this.
In RAM, however, bits are comprised of electrons (or lack thereof) and they do have a mass which is about 9.10938215 × 10−31 kg. So for a GiB of memory, assuming equal distribution for zero and one bits, we get around
4294967296 n × 9.10938215 × 10−31 kg
4294967296 would be the number of one bits in memory (assumed to be 50 %) and n would be the number of electrons that are on average in one bit. I have found one source1 that specified this number at around 105.
So we can give an estimate of how much mass 1 GiB (or 1 GB) of memory would have:
1 GiB, half filled with ones ≈ 3.91 × 10−16 kg = 391 femtograms
1 GiB, completely filled with ones ≈ 7.82 × 10-16 kg = 782 femtograms
1 GB, half filled with ones ≈ 3.64 × 10−16 kg = 364 femtograms
1 GB, completely filled with ones ≈ 7.29 × 10−16 kg = 729 femtograms
So in general you can assume that weight to be pretty unnoticeable (or, with hard disks to be downright nonexistant).
1 These lecture slides, but they are in German.