The Teshoo lama starts out from the very first chapter of Kim on a personal quest. He explains to the curator of the Wonder House that the Buddha once took part in a test of his archery skill. His arrow flew so far that it passed the furthest target, and where it landed, a river sprung up. If the lama can bathe in that River of the Arrow, it will cleanse him of sin and he will be Enlightened.
But here's the thing: there is no River of the Arrow. Or at least, there are tons of archery contests in folktales and popular culture (heck, we just saw one in Brave). And the Buddha also preaches a Parable of the Arrow: he gives an example of a man who, having just been shot by a poisoned arrow, wants to stop before he gets any treatment for his horrible wound to find out who shot the arrow, what clan he belongs to, and all of these details.
But none of these answers are going to heal the giant, bleeding arrow wound in his body. Similarly (for the Buddha), skeptics who demand answers to questions like, "what happens after we die?" and "what is the nature of the human soul?" are missing the point—no one will learn the answers to these questions before they die. The point of Buddhist practice is to reduce suffering—to treat the arrow wound, instead of figuring out all of these less immediate details about how it happened in the first place.
So, there are archery contests and there are Buddhist teachings that involve arrows, but there is no test of strength for the historical Buddha, and there is no River of the Arrow in popular Buddhist belief. Since Kipling actually bothers to mention the real live Four Holy Sites of Buddhism in India by name (1.54), we know that he is capable of decent research. So why did he make up this story of the River of the Arrow for his extremely learned and faithful lama?
We think that Kipling invented this holy object for the lama's quest precisely because it is so individual and personal. The lama goes to all of the sites sacred to Buddhism in India during the novel, but even though he talks with many learned people—the Jain monks in Benares and the curator of the Wonder House, to give some examples—they can never help him find precisely what he is looking for.
The lama's holy pilgrimage is not for something that's important to all Buddhists—the River represents his personal Enlightenment. And because it's personal, the River of the Arrow does not need to be a popularly recognized sacred river like the Ganges or the River Jordan. It just has to be significant to the lama.
The River of the Arrow turns out to be an apparently insignificant brook near the Kulu woman's home. The lama stumbles into this River while he is in a trance brought on by intense fasting. So the River is sacred to the lama because it was in the right place at the right time for him to fall into it just as he was reaching Enlightenment. The River appears just as the lama's quest comes to an end, so it is the proof of his religious and spiritual achievement.