The gift as I understood it was the language itself, which if I'm not mistake would allow them to think like the aliens and see time the same way as they did. This in turn unifies the entire planet in the future and supposedly it will develop earth's civilization advanced enough that in 3000 years they will help the aliens in some way.

Foreshadowed by the different translations of war in Sanskrit

That's correct, the weapon was their language, "using" it implied sharing their knowledge with the other 11 countries to unlock perfect communication. It was a mistranslation- "tool" or "gift" would have been more appropriate for the sentiment they were attemping to get across, and that mistake in tranlating a single word nearly doomed both of our species. Which goes to show the importance of language.
thengamer.com

Pretty sure the gift was world peace. The usa defeated those commie warmongering china and russia! Wait you said china and russia have started how many wars and invdaded how many countries?? Oh well in that case: dey took our durr!

Yeah it was unobvious that it was actually trying to say "we bring gifts" bc that is the easiest thing to miscommunicate for a good drama

She said weapon could mean tool. Although later I got that it meant tool of language. They weren't fluent in earth's spoken languages after all.