Article in your name an incredible story (4)
I believe the answer is:
yarn
'an incredible story' is the definition.
(I've seen this in another clue)
'article in your name' is the wordplay.
'article' becomes 'a' (the indefinite article in English).
'in' means one lot of letters goes inside another.
'your' becomes 'yr' (shortened version of 'your' in eg textspeak).
'name' becomes 'n' (common abbreviation e.g. on forms).
'a' put within 'yr' is 'yar'.
'yar'+'n'='YARN'
(Other definitions for yarn that I've seen before include "Rambling tale" , "that'll keep you in stitches?" , "Sailor's tale" , "Spun thread or wool or tall tale" , "Sham" .)