A graviton stabilizer was a component in the artificial gravity generator. This stabilizer modulated the level of gravity produced, and if it failed, it could pull too hard and make a person feel ten kilograms heavier.
In 2375, the USS Defiant's graviton stabilizer was damaged, and Captain Sisko gave Chief O'Brien three days to fix it. O'Brien then gave the responsibility of finding a new stabilizer to Ensign Nog.
While the USS Sentinel was visiting Deep Space 9 about the same time, they had a surplus graviton stabilizer. Nog procured this in a series of trades. (DS9: "Treachery, Faith and the Great River")