The most common and most likely answer to this crossword clue is the 3 plus 5 letter word SEA ONION
S
E
A
O
N
I
O
N
Buddy explains!
The straight part
We know that "Lily of the Mediterranean" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lily of the mediterranean = sea onion.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.