The most common and most likely answer to this crossword clue is the 5 letter word NAOMI
I've seen this crossword clue in The Times
N
A
O
M
I
More clues leading to the result NAOMI
- Single chap from east embracing old woman
- Woman I grumble about
- Female writer briefly including a mystic syllable
- Biblical character I whine about
- Biblical girl I make complaint about
- Girl reversing in limo anxiously
- Girl raised grouse on island
- In my opinion, an upstanding woman
- Miss, possibly, I lament on reflection
- One complains to the top woman
Buddy explains!

The straight part
We know that "Miss, possibly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case miss, possibly = naomi.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.