Tool to pick up name on inside of clothing (5)
I believe the answer is:
tongs
'tool to pick up' is the definition.
'name on inside of clothing' is the wordplay.
'name' becomes 'n' (common abbreviation e.g. on forms).
'on inside of' is an insertion indicator.
'clothing' becomes 'togs' (togs is a kind of clothing).
'n' put within 'togs' is 'TONGS'.
(Other definitions for tongs that I've seen before include "Grasping tool; Chinese secret societies" , "Fireside instrument with two gripping arms" , "Holding device" , "Item of tableware" , "Devices used for gripping objects" .)