"... he who has furthest to travel will go fastest with willow."
-- Ollivander family motto (Pm)
Willow is a tree which produces wand wood (Pm).
Willow wood often chooses an owner with a hidden insecurity, but with great potential. This wood has healing powers, and is also good for non-verbal spells (Pm).
Lily Evans Potter's wand was willow, 10 1/4", swishy, and "nice for charm work" according to Mr. Ollivander (PS5) Her wand does not correspond to the Celtic Wand Wood Calendar, which has rowan as the tree for January 30 (JKR).
Commentary
Notes
Willow trees actually do have healing powers hidden within their very bark. The genus Salix contains "salicin" or salicylic acid, used as a painkiller for centuries, and the basis for commercial aspirin and creams for sore muscles. source: Britannica
Weeping willows are a symbol of death and mourning, and often depicted on gravestones, or planted near graves. So perhaps Lily's wand wood foreshadowed an untimely death, and represents what her loss meant to Harry.