Changing her name due to meeting other girls with the same one feels like it’d be unnecessary and complicated—in part due to her age. Since she’s old enough to respond to her name, I agree that it would indeed be a very hard switch for her.
Perhaps she could be [name_f]Lily[/name_f] [name_f]Alexandra[/name_f], used as a double name! (I loooove this idea myself. I know a little girl who exclusively goes by her first name and middle name together, to distinguish herself!) If that’s a mouthful, perhaps [name_f]Lily[/name_f] + a nickname for her middle name.
Ideas in this vein, using nicknames for her middle:
[name_f]Lily[/name_f] [name_u]Alex[/name_u]
[name_f]Lily[/name_f] [name_f]Andra[/name_f]
[name_f]Lily[/name_f] [name_f]Lexa[/name_f]
[name_f]Lily[/name_f] [name_f]Alexa[/name_f]
[name_f]Lily[/name_f] [name_u]Lex[/name_u]
[name_f]Lily[/name_f] [name_f]Alla[/name_f]
[name_f]Lily[/name_f] [name_f]Leanne[/name_f]
[name_f]Lily[/name_f] [name_f]Ann[/name_f]
[name_f]Lily[/name_f] [name_u]Allie[/name_u]
[name_f]Lily[/name_f] [name_f]Anna[/name_f]
[name_f]Lily[/name_f] [name_f]Xandra[/name_f]
[name_f]Lily[/name_f] [name_u]Andy[/name_u]