The most common and most likely answer to this crossword clue is the 3 letter word GAB
I've seen this crossword clue in The Times
G
A
B
More clues leading to the result GAB
Buddy explains!

The straight part
We know that "Rabbit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rabbit = gab.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 GAB mean?
GAB noun- Idle chatter.
- The mouth or gob.
- One of the open-forked ends of rods controlling reversing in early steam engines.
- To jest; to tell lies in jest; exaggerate; lie.
- To talk or chatter a lot, usually on trivial subjects.
- To speak or tell falsely.