The most common and most likely answer to this crossword clue is the 4 letter word TWIT
I've seen this crossword clue in The Guardian
T
W
I
T
More clues leading to the result TWIT
Buddy explains!

The straight part
We know that "Idiot" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case idiot = twit.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.
What does TWIT mean?
TWIT noun- A reproach, gibe or taunt.
- A foolish or annoying person.
- A euphemism for "twat", a contemptible or stupid person.
- A person who twitters, i.e. chatters inanely (see usage notes).
- To reproach, blame; to ridicule or tease.
- To ignore or killfile (a user on a bulletin board system).