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From the script:

LEGOLAS:
This is no mere ranger. He is Aragorn, son of Arathorn. You owe him your allegiance.

BOROMIR:
Aragorn? This... is Isildur's heir?

LEGOLAS:
And heir to the throne of Gondor.

ARAGORN:
Havo dad Legolas (Sit down Legolas)

BOROMIR:
Gondor has no king. Gondor needs no king.

Well, he is Aragorn II. So, maybe (and it's a big maybe) the name Aragorn did ring a bell. Or, as Stewards, they were aware of the names of the kings. (Again, just supposition) So, they knew that Arathorn had a son named Aragorn. They are familiar with the name, but not the face.

Just finished reading the section about the Rangers in the Appendix. They haven't really had any contact with Gondor since the fall of Arthedain, over a thousand years ago. I think the book makes it real clear that the men of Gondor, with possible exception of sneaky Denethor and his palantir, have completely forgotten that the Rangers are still out there.

When Boromir comes to Rivendell, he's coming to a place that the men of Gondor view as a myth, they've been out of touch with the North for so long.