From Biography Today Magazine, 1994 (in an obituary for River Phoenix):
John and Arlyn [Phoenix] chose for their firstborn a symbolic name, after the river of life in the Hermann Hesse novel, Siddhartha, then a cult classic. Their daughters were named in the same fashion: Rainbow Joan of Arc, called Rain; Libertad Mariposa, translated into Liberty Butterfly and shortened to Libby; and Summer Joy. The family's only other boy, Joaquin Rafael, asked at the age of four to change his name so that he could be like the others. Given permission by his father (who was raking leaves at the time), he chose Leaf.
And it seems his mother Arlyn (AKA Heart) also commented on it:
River, Rainbow, Leaf, Liberty, Summer... these are all such beautiful things. We just wanted to remind people of the beauty around them. Leaf even chose his own name, we didn’t give it to him. They can change them anytime they wish. They could change them to Larry, I don’t care.
Obviously, he changed his name back later; according to Wikipedia, it was at the age of 15.