But I can't find a posting dedicated to the origin, so here's one. It claims the phrase goes back to Lewis and Clark:
The History Behind The Phrase 'Don't Be An Indian Giver'
By Lakshmi Gandhi
Author David Wilton argues in his 2004 book Word Myths: Debunking Linguistic Urban Legends that the concept of an "Indian gift" arose when white settlers misinterpreted the Native American concept of bartering:
"To an Indian, the giving of gifts was an extension of this system of trade and a gift was expected to be reciprocated with something of equal value. Europeans, upon encountering this practice, misunderstood it, considering it uncouth and impolite. To them, trade was conducted with money and gifts were freely given with nothing expected in return. So this native practice got a bad reputation among the white colonists of North America and the term eventually became a playground insult."
This definition stuck and the phrase "Indian giver" made its first appearance in linguist John Russell Barlett's Dictionary of Americanisms in 1848.
But the above explanation may be the most common one. I don't know if most people think it began with Lewis and Clark, but most think it refers to a misunderstanding of Native customs.
For more on "Indian giver," see Limbaugh Calls Warren "Squaw Indian Giver" and Lauer Calls Vieira "Indian Giver."