The most common and most likely answer to this crossword clue is the 5 plus 10 letter word SHADY REPUTATION
I've seen this crossword clue in The Times
S
H
A
D
Y
R
E
P
U
T
A
T
I
O
N
Buddy explains!
The straight part
We know that "The Bowery's name?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the bowery's name? = shady reputation.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.