Kairi's Grandma

Kairi's Grandmother only appears in a flashback of Kairi's lost memories. In that memory, she is telling Kairi a fairytale about the light. Sora, upon meeting Kairi in the Underwater Cavern in Traverse Town, acknowledges that there is a deeper meaning to this story, and is quoted saying that it was "more than just a fairytale".

Her name and history remains in the dark, though she is presumably deceased prior to the events of Kingdom Hearts, as she makes no further appearance in the series and is not mentioned after the first game. Her presence in the Library of Hollow Bastion may or may not indicate some family connection to Ansem, as it may simply have been that no other suitable place for the storytelling existed at the time of the game's production.

The Fairytale
The fairytale goes like this:

Long ago, people lived in peace, bathed in the warmth of light.

Everyone loved the light.

Then people began to fight over it.

They wanted the light for themselves.

And darkness was born in their hearts.

The darkness spread, swallowing the light and many people's hearts.

It covered everything, and the world disappeared.

''But small fragments of the light survived... in the hearts of children.''

With these fragments of light, children rebuilt the lost world.

It's the world we live in now.

But the true light sleeps, deep within the darkness.

That's why the worlds are still scattered, divided from each other.

But someday, a door to the innermost light will open.

And the true light will return.

So, listen, child.

Even in the deepest darkness, there will always be light to guide you.

Believe in the light, and the darkness will never defeat you.

Your heart will shine with its power and push the darkness away.

''Do you understand, Kairi?

'''In the KH Manga series, the fairytale (with some modifications) is used as a prologue to the story. It goes like this:'''

Long ago, the world was united and filled with warm light.

People loved the light, and eventually began fighting over it.

Then, Darkness found its way into people's hearts.

Darkness consumed the hearts and light of the people,

and in a flash it spread...

The world disappeared into the darkness.

But a glimmer of light remained in the hearts of children...

Children gathered their glimmers of light and recreated the worlds.

The recreated world, however, was no longer united...

It was divided into several smaller worlds.

Because the true light was still hidden deep within the darkness...

Trivia

 * One line from Kairi's grandmother's story, "Even in the deepest darkness, there will always be light to guide you," is a recurring theme in the Kingdom Hearts series. It could very well be an allusion to Kingdom Hearts, which is situated deep in the Darkness. This line was uttered in different words by Sora, Riku, Naminé and King Mickey. Tetsuya Nomura, according to Director's Secret Report XIII, used this same line for the formation of the final scene of Kingdom Hearts II in Dark Meridian, where the shore was situated in the Realm of In Between whilst the sea was situated in the Realm of Darkness. The Door to Light opening in the water is reminiscent of this line.