It's known that East Asian ships (with their crews alive, dead, or simply vanished) have washed up on the shores of the West Coast of North America for centuries and certainly longer, thanks to the currents in the North Pacific.
Now, let's say that one of these ships happen to survive the journey, and with the ship includes something with Chinese characters written on it. Now, I'm not sure why something like that might be on what is no doubt a smallscale vessel. Now, perhaps one of the fishermen or whoever also survived the journey. Somehow, the local Indians he encounters are able to get from him (once they nurse him back to health) that the paper effectively "talks", even if the man himself can't read it (he probably can't).
Here's where we run into the problem of need. What would these people get out of it that they would need it? The West Coast didn't have agriculture, though it did have populations which were at the limit for how much the lifestyle of the indigenous peoples could support (about 300,000 to a million). It also is a region of incredible linguistic diversity. The best place might be the Pacific Northwest, rather than California, which I believe had more organised civilisations, although still nowhere near the level of organisation seen elsewhere. The Salishan languages native there also have more consistent phonology (which seems to be insanely complex for English speakers, but that's beside the point).
So what happens next? My guess is (for the sake of this topic) this Asian man encounters a certain individual who ends up obsessed with the idea, and this person starts fooling around with it whenever they have the time to. Within a few years, they have a workable script, which is occasionally used here and there (grafitti?). Perhaps it spreads far, in mutated form, to people who have even more need of it, like the Mississippians. To do so, you'd have to cross the trade routes through the Great Plains, meaning anything that reaches there would probably be highly different. I know I mentioned it being the Pacific Northwest, but if it were in Central/Southern California (thus our castaway lands among maybe the Chumash or Tongva), it would be easier to get it to the Ancestral Puebloans in the Southwest, who might also find it useful.
My guess is the actual script would be vaguely Chinese looking, probably resembling most whatever characters were found on the ship. The most common form would be engravings on stone and bark, meaning it would be very square looking and angular. Yes, I'm going for a sort of Sequoyah approach here (and I don't mean the Sequoyah tree), though there's plenty of other interesting examples (Vai, inspired by Sequoyah, also Hmong). It seems these can be as simple or complex as needed to fit the language. The Hmong script in particular seems to have gone through a couple of revisions, which no doubt this one would.
Is this necessarily plausible? Is this a case of the right individuals meeting at the right time and thus changing the world forever?
Any thoughts, discussion? Just something I mostly came up with on the fly.