The most common and most likely answer to this crossword clue is the 8 letter word SLIGHTLY
I've seen this crossword clue in The Sunday Telegraph
S
L
I
G
H
T
L
Y
More clues leading to the result SLIGHTLY
Buddy explains!

The straight part
We know that "a little bit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a little bit = slightly.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 SLIGHTLY mean?
SLIGHTLY adverb- Slenderly; delicately.
- (degree) To a small extent or degree.