The most common and most likely answer to this crossword clue is the 7 letter word AGNOMEN
I've seen this crossword clue in The Times
A
G
N
O
M
E
N
Buddy explains!

The straight part
We know that "Extra name" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extra name = agnomen.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 AGNOMEN mean?
AGNOMEN noun- An additional cognomen given, as an honour, to a Roman citizen.