[Boromir88]

Well, ok so we know he went to Rivendell to get answers to a riddle he had in his dream. Faramir had the same dream and he had it more times. So why not Faramir?

First I will say that it was Denethor and the Elders who decided Boromir should go instead of Faramir:

"Alas that ever he went on that errand! I should have been chosen by my father and the elders, but he put himself forward, as being the older and the hardier, and he would not be stayed."~The Window on the West

So, Boromir was the one chosen to go, but he was the one that put himself in front and said he would go. So, why did Boromir feel the need to go?

I'm gonna say it was fate. Both Boromir and Faramir were meant to come across the Ring and face the "test" of the Ring. Let's consider if it was Faramir chosen to go to Rivendell and join the Fellowhsip and Boromir stayed back at home.

If this situation happened, there are many things that could unfold. First, would Faramir fall to the temptation of the Ring and cause Frodo to leave the Company? If it wasn't for Boromir I don't think Frodo would have ever come across the decision of leaving the company (which was the smartest thing for Frodo to do). He found it hard to abandon the company even AFTER Boromir tried to take the ring, so I find it hard that Frodo would find the need to leave the company if he didn't have someone (like Boromir) to sort of give him an unintentional nudge. Would Faramir fall to the temptation of the Ring? Whether he would have or not, is up to you. But, Frodo came up with the decision to leave the Fellowship because of Boromir trying to take the Ring, and if he didn't have that happen then Frodo would not have left the company.

Another possibility is it would create a problem when Aragorn goes to Gondor. Assuming Boromir would still be alive when Aragorn gets to Gondor it would create strife and perhaps civil war. At first meeting Boromir was very apprehensive to accept Aragorn as the "heir," and King. But after their long time spent together in the Fellowship as Frodo remarks he always "showed Aragorn with honor." Boromir grew to accept Aragorn and they became rather good friends, and when Boromir dies he accepts Aragorn and tells him to save his city. So, if Boromir had stayed in Gondor, and Aragorn arrived, Boromir would have thought like he did when they first met in Rivendell, who in the heck is this guy? We don't need you, and then starts the trouble. In BoLT Tolkien contemplated this very idea of Boromir becoming evil and joining Saruman to try to overthrow Aragorn's claim:

Boromir and Aragorn (who notes a change in Boromir - who is keen to break off the chase and go home) reach Minas Tirith, which is besieged by Sauron except at back? Siege is briefly told from point of view of watchers on battlements. Evil has now hold of Boromir who is jealous of Aragorn. The Lord of Minas Tirith is slain (9) and they choose Aragorn. Boromir deserts and sneaks off to Saruman, to get his help in becoming Lord of Minas Tirith.~The Story Foreseen from Moria, The Treason of Isengard

So, had Boromir stayed behind in Minas Tirith, Aragorn would have faced strong resistance to him becoming King. But, since Boromir went to Rivendell and joined the Fellowship, he was able to grow and accept Aragorn.

The last other possibility, and probably the scariest of all. Is had Faramir and Boromir been switched. Assuming Frodo would be captured in Ithilien, it would have been Boromir faced with the decision about what to do with Frodo. Let him go on his quest to destroy the Ring, or bring it back to pops. I think we all would know what would have happened in this situation.

So, in conclusion:

1) He was chosen to go

2) The best thing to happen, was strangely enough, to have Boromir go to Rivendell instead of Faramir. As there would be a wide range of possibilities and changes that could have happened had Boromir stayed in Minas Tirith.

Edit:

It might have been his way to assert his superiority over his brother or to gain favours from his father... or both.~Farael

I don't think it was either. Despite Denethor's favoritism to Boromir the two brothers didn't let that come between them:

Yet between the brothers there was great love, and had been since childhood, when Boromir was the helper and protector of Faramir. No jealousy or rivalry had arisen between them since, for their father's favour or for the praise of men. It did not seem possible to Faramir that any one in Gondor could rival Boromir, heir of Denethor, Captain of the White Tower, and of like mind was Boromir.~Appendix A: The Stewards

Even if Boromir was the favoured one and one that got more praise from the "Men," they didn't let that get between them. This may even bring up another possibility. Perhaps, Boromir felt as if he would be protecting Faramir, in not having him embark on such a long and tedious journey.