How does Lily’s quest to find out about her mother turn into a quest to discover her inner self?

Lily, as you note, starts out to find out about her
mother, but her journey results in self-knowledge.  There are many reasons this
happens.

First, Lily's journey ends in a place where her
mother has been known and loved, as an adult and also as a child.  This gives Lily
knowledge about her mother that she would otherwise never have had.  Since we are all
part of our parents, in a figurative and literal sense, our knowing and understanding of
them is an important part of our knowledge and understanding of ourselves.  August and
her sisters are able to fill in the blanks for Lily, about her mother as a child and
what happened to her mother when she left T-Ray.  This more complete picture of Deborah
helps Lily to see her mother as a human being with her own needs and frailties, which is
important because part of growing up is being able to see our parents as individuals,
not just as our parents.  This also helps Lily to accept what happened to her mother and
allows her to gain some acceptance of herself. Now that she understands what led up to
the day her mother died, she is able to forgive
herself.

Second, Lily's journey exposes her to a different
place, different people, different experiences, and different ideas.  This allows her to
examine her own ideas and feelings with a new perspective.  She learns about community,
from the Boatwrights, the bees, and the Daughters of Mary.  She learns about the grace
and beauty of hard work.  She  learns that people have problems and flaws, but they can
manage to love one another, as she sees how May, with all her difficulties, is loved and
accepted by her sisters, and how June, who is a very prickly sort of woman, is loved and
accepted.  This allows her to accept her mother and to accept herself.  She gains
insight into her previous ideas about race, which allows her to grow into a person
without prejudice.  She also gains some insight into T-Ray, and this is important
because Lily really needs that in order to move on, to not be crippled by a hatred of
him.  She falls in love, which is an important part of growing, and in this case, also
helps her to grow as a human being without prejudices.  She is exposed, for what appears
to be the first time, to spirituality, in the form of Black Mary, which helps her to
understand that there is a power larger than she, which also helps her to gain
acceptance of herself and others.

The Secret Life
of Bees is a fine example of the literary tradition of the "quest."  Lily
begins with a great deal of neediness, guilt, resentment, and heedless prejudice, and by
the end of the book, she has gained an understanding of herself and the world around her
that will allow her to continue to grow and thrive.  I do wish Sue Monk Kidd would write
a sequel because I think we would all love to know what happens to Lily
now.