I think we should probably forget New York. It's far, far beyond the reach of the French.
I agree that an obvious bit would be Hudson's Bay Company, but am less certain the French would end up with it. They simply couldn't take and hold the place, so they'd have to be given it in the peace deal. That begs the question of why they'd want to go after such a big tract of land they couldn't well defend when positions in the Caribbean, Gulf of Mexico, Africa, India, and possibly even Europe would be on the table.
What they could accomplish during the war would be successes in the borderlands. The Iroquois could have been crushed, the western Appalachians secured, possibly positions taken and held in Nova Scotia / Acadia. The latter - which still had many French settlers in the OTL New Brunswick area - would be the region of greatest long-term significance to change hands (if it did).
It's not that I disbelieve that the HBC lands or Newfoundland would switch, but they wouldn't do so during the war. The British had a bigger navy, so both areas were safe. Wound the British navy and they're still safe, because the French will concentrate their ships to attack English positions almost everywhere before North America. They could change hands at the peace table, but then you beg the question: Why take that when you can take something warmer, richer, and more defensible?
Still and all, significant effects. A generation could see the French presence in Vermont and Acadia increase substantially, outposts like Detroit develop, and a permanent presence be made in OTL Ontario.
Now if you want French conquest of the colonies.... First you need French conquest of Britain.