My father has an aunt called Millie but her actual name is Melanie so there is another option and if your daughter didn't like Millie, she could be called Mel or Lanie.

My Bestfriends Niece Is Called Millianna
And I Also Know A Girl Called Milly-Rose
Its A Cute Name

There are alot of people who seem to criticise this name but personally I think its a lovely name. Its 1st on our top 2 names!! We know a baby called Millie so its not uncommon, and personally I dont think of it as an olden name. I would have said that about Alfie before for a boy until my mate called his son Alfie, now I think it really suits a boy. So much so that we wanna call our boy Alfie but cant cause of my mate's son (damn).