Precious case for duchess's customary country wear (6)
I believe the answer is:
tweeds
'customary country wear' is the definition.
(type of clothing worn in the countryside)
'precious case for duchess's' is the wordplay.
'precious' becomes 'twee' (I've seen this before).
'case for' means to remove the middle letters (the letters which encase the others are used).
'duchesss' with its centre removed is 'ds'.
'twee'+'ds'='TWEEDS'
(Other definitions for tweeds that I've seen before include "Country wear" , "clothing items" , "Sensible clothing" , "garments" , "Country clothes" .)