Either the nut slot is too big or there isn't enough tilt over it, causing the string to vibrate in the slot as opposed to vibrating after it only, and the friction damps the whole string motion, or the same happens in the bridge, perhaps a bridge part is loose and vibrates along with the string instead of being a steady suspension point, which effectively kills the sound/vibration. Brand new strings might take a short while to set and thus sound OK in the very beginning.
Check the nut/tilt over it. Rectify with string trees or fill the slot a little with superglue until it's nice and tight. Check and/or replace any loose parts in the bridge (the actual bridge piece or height adjustment screws.
Can't think of anything else that could do that. No, wait...do you have very high polepieces in mics under G string? Magnetic pull cold cause it as well.