I would go about it the opposite way. If I were to meet someone from the future, what would I ask to prove it? Off the top of my head I would ask for the solution to some great mathematical mystery. “How are the prime numbers arranged? How do you combine general relativity with quantum mechanics? Is P != NP? Is there a solution for a^3 + b^3=c^3?...” I’m certain no one on earth can provide the answer now, but I’m pretty sure they are all solvable. Be careful with time travel that the people you meet are aware of such thing and don’t have concepts like religion or soothsayers. If you predict the future in a society like they would likely not understand. “This guy said Caesar would be murdered on the ides of March, then it happened!” “Wow, he must be the son of Jupiter or something... what’s a “timetraveler?””
Personally, if I was thrown back in time a few centuries, I would do everything in my power to avoid revealing I'm from the future. I'd already have a hard time with accent, lingos, etc. But as others have mentioned, we then develop the issues of Paradoxes, where even my presence will chance the timeline and the future I knew may not be there for me if I am able to return. So I would do my best to leave as little trace of myself as possible, and keep as isolated as possible till I return to what will hopefully be my present. Just remember the Prime Detective doesn't apply to primitive alien cultures.
Use language from the character's original time period, say if the character is from the 60s and goes forward to 2020 or further - you could use dated vocabulary and slang.
Thank you for your contributions. Jerseyrunner: Good advice. I thought about it. I actually tried to search the types of time travel stories that existed before the end of the civil war. I found a novel (the name escapes me right now) about a time traveler that comes back with a bunch of letters from 1997 and 1998. The only eras my time travelers visit (so far) are civil war and 2020s through 2030s.
I'd say your MC's first task should be to convince her past self of who she is. (How far back in time does she have to go? Is it even possible to meet her past self?) Once that's covered, she can use her PS to then go to the people she really wants to recruit with information like what her future self knows about what will happen to her in the upcoming days/months. What if the MC's past self is also essential to her task? Try considering that and see if it can work with the rest of your story. Good luck!