Native peoples overwhelmingly preferred to fight for the British instead of the Americans for the simple reason that the British were, by that time, much more likely to be receptive to the needs and wants of Native populations. The Americans, on the other hand, had recently been nothing but trouble.
But we need some background, especially considering that British Indian policy was pretty abhorrent until after the French and Indian War. First, we have to look at the French and Indian War and the contrasting styles of colonization evident by both the British and the French, and how it related to Native peoples.
British colonization is fairly well-known. It incentivized individuals to obtain and develop their own tracts of land, which put a lot of pressure on their colonies as a whole to expand westward. The key here is that the British, viewed from a very distant, non-individual way, wanted to obtain, control, and develop land. Problem with this, of course, is that Native peoples occupy that land. This led to a lot of conflict, both in peaceful terms - negotiating treaties, making alliances and promises, and specie/goods payment in exchange for land - and non-peaceful - all the wars.
This method of colonization encouraged whole families to emigrate, which is a critical point I'll address later.
The French, on the other hand, were less interested in land than they were interested in controlling and influencing trade networks. The primary means of making money on a colonial level in North America was, prior to about 1830, the fur trade. The way the fur trade worked, again at a colonial level, was by making Natives a cooperative labor force (rather than a coerced labor force, like in South American colonies). They would harvest the pelts during winter, and trade them to French traders for non-perishable goods like wool (especially colorful wool), trade muskets, iron and steel goods like pots, pans, and knives, and wampum. This type of relationship meant that the French were less interested in dominating land and more interested in having control of certain trade nodes, like the Straits of Mackinac, the Detroit river, and the Niagara region.
This meant that the French had a huge incentive to form and maintain friendships with influential Natives. The fact that there were mostly French men in the new world (instead of whole families) meant that many of the fur traders and even coureur de bois were young, single men. Natives were keen to form kinship relations with fur traders, because of the access to European goods it gave them. So there were a huge number of marriages between Native women and French men, orders of magnitude more than between Englishmen and Native women.
This also meant that there was a massive difference between the population of New France and New England.
Even with the somewhat decentralized approach to colonization, land was important. Native peoples had cultivated many hunting grounds, encouraging the kind of fruit and vegetable plants that attracted deer and beaver and the like through controlled burning and other methods, which is in part what made them so good at harvesting pelts. But with the English constantly encroaching, it was very difficult to preserve hunting grounds in this way.
So the French and Indian War. The French attempted to claim all the land west and north of the English colonies by planting little metal plaques that backdated their claim to the land. The English countered by establishing border forts on land claimed by their authorities, and attempted to backdate their own claims with surveys and land titles. They eventually sent out a young and somewhat oafish George Washington to do some of the surveying, and it became a big disaster when his native guides took advantage of his naivety and attacked a French party.
The war rages on in America and Europe for a long time, slowing down around 1761. The British take over a bunch of French fur trading posts, including Michilimackinac, one of the most strategically important fur posts in North America, as it controls the most prominent gateway from the east to the west, and straddled the nexus of most of the waterways in North America.
The British replace the French policies of gift-giving to Native groups with an attempt at tough love. They expect that the Natives are just going to roll over and take it, but instead, in 1763, a Delware named Neolin starts preaching a pan-Indian religious philosophy and a rejection of European goods and lifeways. An Odawa warrior named Pontiac takes the lead, and is the snowball on a ski slope that kickstarts a massive Great Lakes centered war against the British. Targeting the most influential forts along the Great Lakes, the pan-Indian forces capture all but Detroit and Fort Pitt (if you really pull my leg I'll tell you about how they captured Fort Michilimackinac because it's a great story but a little too long for this post).
Pontiac failed to capture all of the forts, but instead, he influenced the British government to completely overhaul their Indian policy. They replaced the gift-giving aspect, sought to make friendly relations with the Great Lakes peoples, and most importantly, instituted the Proclamation of 1763.
The Proclamation's biggest controversy was that the British crown made all colonial settlement west of the Appalachian Mountains illegal. There were already thousands of Englishmen living beyond that line, and the prospect of entire industries - especially the land speculation industry - suddenly drying up was a major factor in increasing anger and resentment by American colonists toward the crown.
Americans had already gone out of their way to show themselves as cruel during Pontiac's War (not to mention during the French and Indian War itself) with the so-called "Paxton Boys." To make a long, horrible story palatable, the Paxton Boys lashed out at unambiguously peaceful, christian groups of local Indians as soon as they heard word of Pontiac's War. They went so far, at one point, as dragging women and children out from town jails (where they'd been placed for their own safety), murdering them and scalping them in front of the entire town.
After Pontiac's War, this stuff didn't stop, even if the Paxton Boys themselves were.
And so in the ten years between Pontiac's War and the War for Independence, the British, who completely overhauled their Indian policy to be more favorable to Indians, remember, are suddenly the targets of land hungry, murderous brutes whose attitude toward land and alliance is utterly anathema to Indian ways of life. The British, in short, are by a wide margin the lesser of two evils.
It helped that, once again, a spirit of pan-Indian unity ignited the frontier, and leaders like Joseph Brant used the memory of Neolin and Pontiac as a way to unify the tribes and nations of the interior to fight against the Americans. Frontier warfare in the War for Independence is sadly overlooked, but it was fed by personal feuds and was astonishingly brutal. Gregory Dowd argues in A Spirited Resistance that the War for Independence was the most successful and unified resistance effort on the part of North American Indians in history, eclipsing even that of Tenskwatawa and Tecumseh in the War of 1812.
There are many forces moving around during the War for Independence: pan-Indian movements are preaching a sovereignty-centered political outlook for Native peoples, the British are keen to dominate the fur trade through fostering and keeping good relations with Natives through personal relationships and accountability, the Americans are going out of their way to be particularly horrible to Natives because their entire domestic, political and economic existence is based on domination of land.
Of course inter-tribal rivalries, old nemeses, bad politics and history between Native groups motivates quite a few to take the side of the Americans. This is based on the belief that not only can they hurt rival clans and bands, but they'll be in a favorable political situation when the war ends, and can expand into rival Indian territory. But those who sided with the Americans are absolutely in the minority.
Fur Trade:
Fur, Fortune and Empire, by Eric Jay Dolin
Facing East from Indian Country, by Daniel Richter (fabulous book I recommend if you have any interest at all in Indian history)
The Middle Ground by Richard White
Pan-Indian movements:
A Spirited Resistance by Gregory Dowd
Beyond Pontiac's Shadow by Keith Widder
The Gods of Prophetstown by Adam Jornter (later than the War for Independence, but it does a good job sketching the political situation resulting from it)
Proclamation of 1763:
A Scratch of the Pen by Colin Galloway