[Farael]

I believe it's a matter of pride... HE was the eldest son so HE was "entitled" to go to Rivendel. It might have been his way to assert his superiority over his brother, or to gain favours from his father... or both.

Maybe it was also Denethor's idea. We know that he played favourites with his sons and that he regarded Faramir as a "wizard's pupil". Knowing that Gandalf may have been at Rivendel, he might have persuaded Boromir to go instead so that Gondor would have been represented by someone who thought along Denethor's lines.