| I see two possibles here. First is that the string is clattering against the fretboard; second is that it's making something else rattle sympathetically.
I recommend you try the following things, from least invasive/expensive to most:
* Make sure the string is tuned to pitch. A detuned string has less tension/more "slack", and can rattle more easily against the fretboard.
* Make sure you are plucking across the string, not down into it. Plucking downwards produces a "clack" when done too hard, and also induces a vibration that is more perpendicular to the fretboard.
* Check the action of that string. It may need to be raised slightly, even if its exactly where the manufacturer says it should be.
* Make sure all tuning machines, bridge saddles, etc are firmly fastened. Slack all strings and tighten the nuts on top of the peghead for each machine. Don't overtighten or you can damage the instrument.
* Check the bridge saddles; if they're the type that has two screws on either side of the saddle to determine string height, make sure both screws are firmly contacting the baseplate. Also, sometimes the bridge saddles can actually rattle against each other from the very slight movement of the saddle because of the vibrating string. Try separating the saddles slightly, or putting a drop of rubber cement in between the saddles (it won't hurt the saddles, and if done cleanly will be both invisible and easy to remove) to act as a shock absorber.
* Check the pots and jack, and make sure they're securely fastened to the control plate/pickguard/body. Is there anything else in the control cavity that could be rattling?
* Check the action of the neck; it may have become slightly back-bowed over time, and this will affect the higher two strings (which are supposed to be closer to the fretboard) more and earlier than the lower two. It may also have too much relief, and if you have been compensating by lowering the saddles, your strings are going to buzz against the frets at the heel. The neck should have the most barely perceptible amount of relief.
* Whether the action looks good or not, make sure the truss rod is at least snugly tightened. Also, if it is already tightened somewhat, try giving it a tweak of about an eighth turn in either direction. I've had some basses where the truss rod rattles sympathetically with the instrument. This normally indicates a poorly-cut truss rod trough, but can sometimes be worked around to your satisfaction by a small adjustment in tension.
* If you cannot tighten the truss rod, that indicates a serious problem. Either the rod's anchor at the other side has come loose from its mountings, or the truss rod itself has broken. Either way, only a competent luthier will be able to fix it, and it is likely to be expensive.
Beyond that, I can't think of anything you could do without seriously tearing the bass apart. | 
