\version "2.11.39"
#(ly:set-option 'point-and-click #f)

\header { title="Showing instrument name with cue notes"}

sopM = \relative c'' { \repeat unfold 2 { c,4 c g' g | d d e e | f f g g | a a b, b | c c d d | } }
sopL = \lyricmode { A B C D E F G H I J K L M N O P Q R S T A B C D E F G H I J K L M N O P Q R S T }
\addQuote sop \sopM
% \addInstrumentNameToQuote "sop" "Sop." % <== this does not exist, but is what I want!

otherVoice = \relative c {
    R1 |
    \cueDuring #"sop" #UP { R1 } |
    a4 a g2 |
    R1*4 |
    \cueDuring #"sop" #UP { R1 | r2 r4 } c4 |
    c8 c8[( b8] a8) g2 \bar"|."
}
otherLyrics = \lyricmode { a b c d e f g h i }

\score {
    \context Staff = "Soprano" <<
        \context Voice = sopV { \sopM }
        \context Lyrics = sopL \lyricsto "sopV" \sopL
    >>
    \header { piece="Soprano Score" }
}
\score {
    \context Staff = "Instrument" <<
        \context Voice = otherV { \clef "bass" \otherVoice }
        \context Lyrics = otherL \lyricsto "otherV" \otherLyrics
    >>
    \header { piece="Instrumental Score" }
}

\score {
\context StaffGroup = "FullScoreStaffGroup" <<
    \context Staff = "Soprano" <<
        \context Voice = sopV { \sopM }
        \context Lyrics = sopL \lyricsto "sopV" \sopL
        \set Staff.instrumentName = "Sopran"
        \set Staff.shortInstrumentName = "Sop."
    >>
    \context Staff = "Instrument" <<
        \context Voice = otherV { \clef "bass" \killCues \otherVoice }
        \context Lyrics = otherL \lyricsto "otherV" \otherLyrics
        \set Staff.instrumentName = "Instrument"
        \set Staff.shortInstrumentName = "Instr."
    >>
>>
\header { piece="Full Score" }
}