First of all, I doubt she would plainly say "I am from Mesopotamia and I have traveled in time", simply because those are concepts we have formalized. She will probably be scared by all these weird looking creatures popping out of horseless and smelly chariots, with funny body odors and body decorations.

That aside, there are several indicators which can corroborate the suspect: language is probably the best one. We don't speak those old and extinct languages, but thanks to the work of many philologists have a decent hunch at how they might have sounded. If this person was able to fluently speak it, and maybe even read the inscriptions, the possibility would be either that she is an academic of the field or something else. And the more academic institutions would fail to recognize her, the more that something else would become plausible.

Then probably a stool sample taken immediately after her arrival would tell that her most recent meal was not from our times, and also a chemical analysis of her hair would probably reveal level of pollutants different from the one we generally experience today: for one, copper extraction gave out a lot of arsenic in the surrounding.

DNA might show some old traits, but it's hard to assess their statistic relevance from a single sample.

First of all, I doubt she would plainly say "I am from Mesopotamia and I have traveled in time", simply because those are concepts we have formalized. She will probably be scared by all these weird looking creatures popping out of horseless and smelly chariots, with funny body odors and body decorations.

That aside, there are several indicators which can corroborate the suspect: language is probably the best one. We don't speak those old and extinct languages, but thanks to the work of many philologists have a decent hunch at how they might have sounded. If this person was able to fluently speak it, and maybe even read the inscriptions, the possibility would be either that she is an academic of the field or something else. And the more academic institutions would fail to recognize her, the more that something else would become plausible.

Then probably a stool sample taken immediately after her arrival would tell that her most recent meal was not from our times, and also a chemical analysis of her hair would probably reveal level of pollutants different from the one we generally experience today: for one, copper extraction gave out a lot of arsenic in the surrounding.

DNA might show some old traits, but it's hard to assess their statistic relevance from a single sample.

Anon in the comment poses also a legitimate question, about her radiocarbon apparent age: I see two possibilities for this.

1. time travel doesn't affect nuclear decay: this means her $C_{14}/C_{12}$ ratio is unaffected, maybe a bit lower than what we have today. Maybe that could be also explained as effect of her diet, not sure about the numbers, but her radiocarbon age would not appear "old"
2. time travel does affect nuclear decay, forcing all of the atoms which would have decayed to do it after the jump: she would show a lower $C_{14}/C_{12}$ ratio before she starts eating current food, and she might probably show also some sign or radiation poisoning, due to the burst of radiation caused by the travel.