Big name last to leave institute (4)
I believe the answer is:
star
'big name' is the definition.
(I've seen this in another clue)
'last to leave institute' is the wordplay.
'last to leave' means to remove the last letter.
'institute' becomes 'start' (to institute something is to begin or start it).
'start' with its last letter taken away is 'STAR'.
(Other definitions for star that I've seen before include "Play a leading role" , "Ball of fire" , "A night light?" , "It twinkles" , "See 21" .)