As has been said several times before, this scene doesn't occur in the books1, so what we see in the film is all we have to work with. This means that, unless Peter Jackson decides to weigh in, we have to rely on speculation. This being the case, here's my two cents:
I've always taken Arwen's statement at face value: She's an elf, and from a family who are powerful, even for the already-powerful Elven race. She's not saying "I give up my immortality for Frodo" or anything like that; she's just saying that she wants Frodo to be taken under the same special consideration that Elves enjoy. She is asking for her Elven-ness to protect Frodo the same way it protects her.
You could rephrase it as something like this:
"Whatever makes Elves like myself super-special, and lets us live forever and heal faster than mortals - I want Frodo to have that too, at least until I get him to Rivendell where my dad can fix him up. Make Frodo an honorary, temporary elf so he doesn't die before we can heal his wounds and stuff".
This is obviously a very simplified version of what she is asking for, but I think it captures the essence of the request pretty well. She isn't giving Frodo anything of her own, and she isn't trading herself (or her life) for him (or his life). She's just asking for him to live long enough to receive medical attention.
1 In fact, Arwen doesn't really do much of anything in the books: she sews a flag for Aragorn, then marries him, says some nice stuff to Frodo and the rest of the Fellowship, eventually makes Sam's daughter a Maid of honor at her court, offers Frodo her spot on the ship to the Undying Lands, and finally, she dies. That's about it. She isn't particularly important as anything other than Aragorn's girlfriend/fiancée/wife.