Girl giving her name to a band? (5)
I believe the answer is:
alice
This clue is a quadruple definition!!
'girl' is the first definition.
(girl's nae)
'giving' is a link.
'her' is the second definition.
'name' is the third definition.
'to a band?' is the fourth definition.
(I've seen this before)
(Other definitions for alice that I've seen before include "A Town Like - - (novel/film)" , "Looking-glass girl" , "Wonderland girl; --- B. Toklas" , "Celia changes into a different girl" , "Lewis Carroll character" .)