Many a song has a girl's name (5)
I believe the answer is:
maria
'a girl's name' is the definition.
'many a song' is the wordplay.
'many' becomes 'm' (Roman numeral for a thousand).
'a song' becomes 'aria' (aria is a kind of song).
'm'+'aria'='MARIA'
'has' is the link.
(Other definitions for maria that I've seen before include "Lady" , "West Side Story girl" , "Plains on the moon; girl's name" , "Girl's name; darker moon areas" , "Lunar plains; girl's name" .)