Sage Gautama had received a boon from Varuna, the lord of the waters, that the land around his hermitage was always supplied with water.
When a drought struck the earth, he incurred the wrath of the other sages. They drove a cow, Go, into his fields. Gautama tried to chase her away, the cow fell dead.
He prayed to Shiva to let flow the water of the Ganga, which he wore in his hair, to earth, so that he could bring the cow back to life with this water. Shiva agreed.
So Gautama brought part of the Ganga to earth, that is why this river is called Gautami Ganga. Godavari it is called because Gautama had killed a cow.