Delighted with the name (3)
I believe the answer is:
eli
'the name' is the definition.
'delighted with' is the wordplay.
'with' indicates the answer is hidden within the clue.
'ELI' is hidden within 'delighted'.
(Other definitions for eli that I've seen before include "Old Testament high priest" , "High priest in the Book of Samuel" , "Highest priest at Shiloh" , "Old Testament priest, the prophet Samuel's teacher" , "Priest, teacher of Samuel" .)