Perhaps that really is what happened - young Lord Rodrigo Diaz de Vivar looked over all the horses his godfather had, and chose a white Andalusian that looked spindly. Exasperated, his godfather yelled "Stupid!" Babieca! And in a bit of reverse psychology, the horse and the boy grow up to become the greatest warrior team in Spain's history.
Or maybe the horse was actually named Bavieca and the name got warped. Which isn't anywhere near as good a story.
But the tale of who Rodrigo became with the help of his mighty horse still makes for a great story. He grew up to beome The Cid (c 1040 - 1099), the warrior who conquered Valencia and made it his own fiefdom. Smart and strong, the white horse Babieca was his loyal friend and fellow fighter during these adventures; El Cid's enemies were frightened of him.
The story of El Cid and Babieca - the two are pretty intertwined - can only be flirted with in a little post. Lean more by going to this article on The Cid at Wikipedia, and this essay on Babieca.
Here's a Flickr image of their sculpture in Balboa Park, San Diego. And here is their monument in Burgos, Spain.