What would a time traveler have to do to convince you he's really one?

Is that a trick question? Lincoln.

I think this coming election is pivotal in almost the same way and I would expect him to know who won. Actually, I can tell you who won each election from 1820 to 1840. Or even 1788 to 1840. I am a bit watery on 1844 to 1856, but then I can tell you who wan from then on. But I would ask if the US still existed and, if not, what happened. He would have to be pretty good to make up such a story on the spot.

As for language, I once read Looking Backward by Edward Bellamy. It was written in, I think, 1887. When they were in 1887, the language didn’t strike me in any way, but when the time traveler reported conversations he had had in 2000, it really clanged. People simply didn’t talk that way in 2000. So old language, until you get back to Shakespeare, is reasonably comprehensible, but expect some differences. Hell, my kids speak slightly differently than I do.
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Shakespeare didn’t speak like Shakespeare. The plays were, for the most, in iambic pentameter. Sometimes it even rhymed. Always, it was full of metaphor people did not speak in the least. If Shakespeare wrote in ordinary speech, we would not be quoting him today.

How about “all of the above”? Do the genetic thing, get experts from a variety of fields talk to him. I don’t think a person could fool everyone.

The future time machine is not going to transport Joe Schmoe walking down the street, nor some crank inventor from his basement. Whoever arrives from the future is going to be as carefully recruited and vetted as an astronaut. He’ll have all the answers, and may well have memorized tomorrow’s newspaper.

This is a good tangent - considering the immense implications that someone could do by going into a time machine (killing Hitler, or any US president, or otherwise wreaking havoc on history) - future generations would have to exert immense control over whoever is allowed to use such a machine. Otherwise the consequences for all of them could be unimaginable.