I just spotted this one on Twitter:
The language is all wrong for the Buddha, and this sounds very 20th century, with a strong dose of self-help.
At first the earliest source I could find for this was from 1993, from a book by John Mason called You’re Born an Original, Don’t Die a Copy!, except there the final word is “perseverance” rather than the “persistence” of the quote on Twitter. There’s no attribution given there, and in a later book, Know Your Limits — Then Ignore Them he just refers to it as a “famous old saying.”
So we now have:
“In the confrontation between the stream and the rock, the stream always wins; not through strength, but through perseverance.”
I found that Readers Digest Quotable Quotes attributes this to H. Jackson Brown, again with “perseverance” rather than persistence.
This helped me find many other identical attributions, including one to Brown’s A Father’s Book of Wisdom. In that book, published 1988 by Rutledge Hill, he attributes the saying to “Dad.” So far I haven’t found any instances of the quote before 1988, so this may be our source.
It’s been attributed elsewhere to Thomas Sutcliffe Mort, but the origins are unclear at present.
It’s almost certainly too literary in style to be something from the Pali canon (although a translator can of course add some “polish.”
Most of the Buddha’s references to streams were to do with “crossing the stream” to the farther shore of Awakening, but here’s one quote where he quotes (seemingly with approval) another teacher’s simile, using the mountain stream to represent impermanence:
Just as a river flowing down from the mountains, going far, its current swift, carrying everything with it, so that there is not a moment, an instant, a second where it stands still, but instead it goes & rushes & flows, in the same way, brahmans, the life of human beings is like a river flowing down from the mountains — limited, trifling, of much stress & many despairs. One should touch this [truth] like a sage, do what is skillful, follow the holy life. For one who is born there is no freedom from death.
I’m not familiar with any verse from the Pali canon referring to waters wearing down rocks, but it’s a big canon, and I haven’t read it all…