Ann Turner
Time of the Bison
1987
from the dust jacket:

Scar Boy hated his name. A name given because you cut yourself on a rock was hardly a real name. The young cave dweller knew someday he would have a deep-inside name he liked — but when? And what would it be? The tradition of his clan was you had to earn a name. If he killed a huge bison, he might be called Bison Man. If he rode like the wind on a wild horse, he would be Wind Boy or Horse Rider — but none of these names seemed to fit.

Then one night, asleep in the cave with his clan, Scar Boy dreams of a horse — a great stallion with wise dark eyes. Next morning, at the river, he makes a red clay statue of the horse in his dream. His father is astonished. "We must save it. This is special: No one has made animal pictures like this since Tiger Man died so long ago."

Because of Scar Boy's skill at making pictures, his father takes him to see Painter of Caves, a wise old man, after a long and difficult journey with many dangers. Painter of Caves leads Scar Boy into a dark and mysterious cavern. In that magical secret place he discovers the true meaning of his gift and what his real name will someday be.