The Happy Hunting Grounds appear in the Legends of the Nipmuck People. The Nipmuck, or "fresh water" people, lived on the banks of rivers and lakes. Part of the New England Algonquin, the Nipmuck believed that "Upon entry into the "happy hunting grounds," reunions with loved ones would take place. Ancestors in the Happy Hunting Grounds were believed to have dwelt "in another dimension," rather than "having ceased to exist" at all.
There is some question as to whether the phrase "happy hunting-grounds" was in fact created by a white man, and by one famous author in particular. "Did Native Americans really believe in the "happy hunting grounds?" suggests the phrase is inauthentic. One theory the article poses involves James Fenimore Cooper's most famous novel, The Last of the Mohicans (1826). In the last chapter of the book, this phrase is found:
Why do my brothers mourn? why do my daughters weep? that a young man has gone to the happy hunting-grounds; that a chief has filled his time with honor?
James Fenimore Cooper was one of Barrie's favorite authors. In Peter Pan, Barrie borrows from Stevenson's Treasure Island (1883), so it's clear he was not above appropriating others' works to enhance his own.