Dine with head following name order (8)
I believe the answer is:
neatness
'order' is the definition.
(neatness is a kind of order)
'dine with head following name' is the wordplay.
'dine' becomes 'eat' (eat a meal).
'with' means one lot of letters go next to another.
'head' becomes 'ness' (both can mean a promontory).
'following' says to put letters next to each other.
'name' becomes 'n' (common abbreviation e.g. on forms).
'eat' put next to 'ness' is 'eatness'.
'eatness' after 'n' is 'NEATNESS'.
(Other definitions for neatness that I've seen before include "Apple-pie order" , "well-tended condition" , "simple elegance" , "smart condition" , "Good order" .)