Mother goes to look up a girl's name (5)
I believe the answer is:
maria
'a girl's name' is the definition.
'mother goes to look up' is the wordplay.
'mother' becomes 'ma' (ma can mean mother).
'goes to' is a charade indicator (letters next to each other).
'look' becomes 'air' (something might have a certain air or look).
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'air' back-to-front is 'ria'.
'ma'+'ria'='MARIA'
(Other definitions for maria that I've seen before include "Dark planes on the moon; girl's name" , "- Callas, twentieth century opera singer" , "Heroine of West Side Story" , "Black --, police van" , "Girl's name; darker moon areas" .)