The most common and most likely answer to this crossword clue is the 6 letter word YAMMER
I've seen this crossword clue in The Times
Y
A
M
M
E
R
Buddy explains!

The straight part
We know that "Rabbit, noisily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rabbit, noisily = yammer.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 YAMMER mean?
YAMMER noun- The act or noise of yammering.
- A loud noise.
- One who yammers.
- To complain peevishly.
- To talk loudly and persistently.
- To repeat on and on, usually loudly or in complaint.
- To make an outcry; to clamor.