In short I had a name for her (3)
I believe the answer is:
ida
'her' is the definition.
'in short i had a name for' is the wordplay.
'in short' means to remove the last letter.
'i had' becomes 'i'd' (contraction).
'name for' becomes 'n' (I've seen this in other clues).
'id'+'a'+'n'='idan'
'idan' with its last letter removed is 'IDA'.
(Other definitions for ida that I've seen before include "Greek mount - Gilbertian princess" , "musical princess" , "- Rubinstein, Russian dancer, actress" , "Cretan mount" , "Name of girl or mountain" .)