I've toyed with the idea of New England going independent when Charles II was restored to the English throne, but they weren't quite ready for independence at that point. Not enough people or industry to stand on their own.
Another possibility: If the king had won the English civil war against parliament, you might have seen enough of his opponents flee to New England to make an independent Puritan state there. New England was very pro-Parliament. The question would be whether the colonies could integrate enough people to be viable that early. Probably not.
Another option: Independent conquistador kingdoms. The last of the Pizarro brothers revolted against Spain in the 1540s, after a Spanish decree that reduced the power of conquistadors over the Indians that served them. The rebels controlled Peru for a few years as I recall it. There probably would have been a similar revolt in Mexico, but the Spanish Viceroy there wisely decided to hold off on implementing the changes. If he had been less politically adept, you might have seen both Mexico and Peru in revolt, which would have been considerably more serious for Spain and might have led to longer-term independence for those colonies. That's not the way to bet, though, because the colonies were too vital to Spain, because of the gold and silver. They would have put enormous effort into getting them back. It would have taken another European country intervening or some other threat keeping Spain distracted to keep them from doing whatever it took to retake those colonies. France would be most likely to intervene, but had no bases in the Americas close enough to help and would have had to penetrate a Caribbean dominated by Spain to do anything. An upsurge in the Turkish threat might be an option. I would have to look at what France and Turkey were up to at the time. England, as I recall, was tacitly allied with Spain against France at this time.
Yet another option: A French Huguenot (Protestant) state. In the 1560s, French Huguenots tried settlements in Brazil and in Florida. Both failed, the Florida one due to a Spanish expedition massacring the settlers. Let's say that the French tried in the Chesapeake Bay area instead of Florida, further from Spanish power and less threatening to the Spanish treasure convoys going between Cuba and Florida. They would find small tribes rather than the unified group under Powhatan. Let's say they succeed in founding a French Virginia in the 1560s, which expands rapidly as French Wars of religion drive out French Protestants and many of those refugees end up in the French Virginia colony. By the 1590's, there is a large, flourishing French Huguenot population radiating out from Virginia. Of course that means no Jamestown and a whole slew of other changes that would make US history unrecognizable, but it could easily lead to an independent European state in the Americas long before the American revolution.