Girl gets old, having concealed name (5)
I believe the answer is:
agnes
'girl' is the definition.
'gets old having concealed name' is the wordplay.
'gets old' becomes 'ages' ('age' can be a synonym of 'get old').
'having concealed' indicates putting letters inside.
'name' becomes 'n' (common abbreviation e.g. on forms).
'ages' enclosing 'n' is 'AGNES'.
(Other definitions for agnes that I've seen before include "Woman" , "Miss Grey" , "St --' Eve, poem by Lord Tennyson" , "Lady" , "The Eve of St. - (Keats)" .)