Recently I was perusing wikipedia (through which all things truly are possible), and I came across a bit of actual history that I found absolutely jawdropping.
Evidently in the seventeenth century the Duchy of Courland (currently the southern half of Latvia) under Duke Jacob Kettler had a lovely little empire stretching from the West Indies (where it for a time ruled Tobago) to several fortresses in Africa which exported, according to wikipedia "sugar, tobacco, coffee, cotton, ginger, indigo, rum, cocoa, tortoise shells, tropical birds and their feathers."
Of course, it goes without saying that if I was stuck in the Baltic winters of the seventeenth century (mini-ice age, remember) and had sailing ships and sufficient gunpowder, I'd sail to the Carribean and claim myself an island as well. Lutherans from frigid climes still do much the same thing in the western hemisphere even today, only instead of weaponry and galleons they wield time shares and winnebagos.
Now, more seriously for our purposes, we all know how relatively small European countries (see Denmark), or relatively non-obvious participants in the Atlantic colonization process (see Sweden) were involved with varying degrees of success in colonizing the New World. Most of them of course fell away quickly, trampled by the militarily more powerful states.
My challenge, assignment, prompt, what have you, is to find the most unlikely European state (especially if it was independent for only a nano-second or so) and find a plausible way for it to lay hold of prime New World real estate, with the wildest current-day deviations from actual history. In short, bonus points if everyone in the Western hemisphere is now speaking Basque.
Go!