In The Way to Rainy Mountain, the story of Grandmother Spider and her adopted child is one of the Kiowa myths retold by Momaday, in the same oral tradition that the Kiowa have always passed down their stories and history.

Grandmother Spider is worried about the Sun's child because she knows how powerful he is, as a diving being, and worries that she will not be able to control any aspect of his behavior. His constant disobedience shows how right she was. For instance, he throws a wheel into the sky after warnings from Grandmother Spider, and this ends with him being cut in two pieces, each of which reforms into a "twin" of the other. Grandmother Spider now has two disobedient children to raise.

Her fears that she cannot keep the twins safe and raise them the way she should continue to be confirmed as they play a game which involves throwing their rings into a giant's cave. However, when they are nearly injured or killed by the giant, they remember their adopted mother's words for once and are able to say an incantation she taught them to save themselves. 

Despite Grandmother Spider's fears, the twins are said to have lived long and happy lives, even after her death. Perhaps she taught them well enough after all.

In short, the Spider Grandmother has many worries just as Momaday's own grandmother has many worries.  Specifically, the Spider Grandmother worries about "raising the Sun's child" mostly because Spider Grandmother is part of the natural world while the Sun's child is partially divine. 

In order to understand this idea, it is important to understand a simple version of the Kiowa creation myth.  It begins with a boy born to a Kiowa woman and the Sun.  This makes the boy both human and divine.  Because the boy's mother cannot raise him, Spider Grandmother raises him.  It is during this time that her worries become paramount.  During her protection, the boy is split into two twins through a miracle.  One of the twins disappears into a lake.  By doing this, he becomes part of the natural world.  The other twin changes into the ten bundles and given to the Kiowa as a divine gift. 

As you can see, Spider Grandmother does insure the survival of the split twin sons, but one of them does disappear.  Spider Grandmother's fear, then, was unfounded.  It turns out that her connection with the natural world was important in that the one twin needed to return there.  In this way, Spider Grandmother becomes a representative of all Kiowa grandmothers:  a sign of tribal unity and tribal harmony.