The most common and most likely answer to this crossword clue is the 7 letter word PAPILLA
P
A
P
I
L
L
A
More clues leading to the result PAPILLA
Buddy explains!

The straight part
We know that "Organ" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case organ = papilla.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.
What does PAPILLA mean?
PAPILLA noun- A nipple-like anatomical structure.