The most common and most likely answer to this crossword clue is the 4 plus 4 letter word REED STOP
I've seen this crossword clue in Independent
R
E
E
D
S
T
O
P
Buddy explains!
The straight part
We know that "part of organ (instrument, not a bodily organ)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case part of organ (instrument, not a bodily organ) = reed stop.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.