J.K. Rowling based Harry's, Ron's, and Hermione's wands off of the wand woods correlating to their date of birth and the Celtic tree calendar. However, as Lily Potter's wand is made of willow, not rowan (her Celtic birth tree), J. K. Rowling did not choose this wand based on Lily's birthdate. Like Hagrid's wand wood, it could be based on a symbolic reference to Lily's personality and character.
The willow tree symbolises psychic abilities, love, deep emotions, intuition, and dreaming. Likewise, the wand wood could reflect Lily's character, in particular her deep love of Harry, her son.