Could be two things.

Either used to mute resonant noises from the string above the nut.

Or to glide it on further on the neck to mute notes (keep them from ringing) when playing an extensive tapping lick where u can't mute the string noises with ur hand that easily.

Basically cleans up the sound, and there's no official name for it.

You could call it a string dampener I guess

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