The most common and most likely answer to this crossword clue is the 5 plus 4 letter word THEME TUNE
I've seen this crossword clue in The Times
T
H
E
M
E
T
U
N
E
More clues leading to the result THEME TUNE
Buddy explains!

The straight part
We know that "Familiar melody" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case familiar melody = theme tune.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.