The most common and most likely answer to this crossword clue is the 6 plus 5 letter word FLOWER POWER
I've seen this crossword clue in The Times
F
L
O
W
E
R
P
O
W
E
R
More clues leading to the result FLOWER POWER
Buddy explains!
The straight part
We know that "What might once have inspired peaceful protest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case what might once have inspired peaceful protest = flower power.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.