I use a "inverted" drawer as a support for my mouse and keyboard, the problem is since it's hollow it reverberates a lot, any thumb on it can be heard from outside my room. What can I do or which material can I use to make it reverberate less?
If that drawer really fits your needs, you need weight in contact with it to fully stop it vibrating.
Fill a pillowcase with enough dry beans or rice or similar density but space-compliant bits to fill the drawer, sew shut and place the inverted drawer over it. The filler must almost completely fill the drawer. Don't use packing peanuts - they're too light, not dense enough.
Also run foam weather-stripping around the drawer edge in contact with your table-top to dampen transfer of vibrations both ways.
Sounds like a problem best fixed by getting a proper support rather than patching up a band-aid. At any rate, a sound absorbing mat (usually used to stop bass enclosures from vibrating) glued full-area to the bottom of the drawer (if the mat isn't self-adhesive, use either prescribed glue or contact glue) should likely do the trick. It's sort of heavy semi-deformable foam. Actually not all that unlike mouse mats.