An MP3 of total silence should be total silence, and that is exactly what you get from Lame-for-Audacity.
Q) So where’s that noise coming from?
Audacity does all “mixing” in 32 bit and that includes Exporting.
When audio is encoded as MP3 the bit depth is dropped to 16bit and is subject to the Audacity dither settings (Edit menu > Preferences > Quality).
There are a few ways that you can avoid the dither noise:
Export as 32 bit WAV and encode using an external encoder that does not apply dither.
Temporarily switch off dither in Audacity Preferences,
Set dither to “Rectangle”
Which is the “best” solution?
Option 1 is a good solution provided that the external encoder is of equivalent quality as Lame, though it is perhaps a bit inconvenient.
Option 2 is a good solution when exporting a single 16 bit track - in this case there is no advantage to applying dither.
Option 3 is possibly the most convenient solution as it can be set once and left. It still has the advantages of using dither when downsampling 32bit tracks and/or mixing multiple tracks during Export, though the subjective effectiveness of rectangle dither is possibly not quite as good as triangle or shaped on non-silence.
Option 4 - ignore the noise and use triangle or shaped dither. In music recordings it is rare that there is absolute silence, and total silence is about the only time that the dither noise is evident.
Q) Would it be better if Audacity did not apply dither when Exporting a single 16 bit track?
A) Probably yes - though it is a “special case” that has been discussed before and the developers seem reluctant to change it.