Fingerrest
One word or two?
A fingerrest (finger rest?) is much like a pickguard except that it is elevated from the face of the instrument to provide a place to, you guessed it, rest your fingers.
Fingerrests are usually called pickguards, although there is no need for a guard on these instruments because the strings are so far from the face that a pick wouldn't touch, let alone scratch it.
Some fingerrests are elegantly designed and subtly blend with the instrument.
Some fingerrests are a fashion statement unto themselves:
Gibson is generally credited with the innovation of the elevated fingerrest, and patented the device in 1909.
Personally, I like to spell it as one word, "fingerrest."