The first answer pretty much says it all, but there is one other reason that Cassius chooses Brutus over other well-respected, eloquent senators. Brutus comes from a long line of noble Romans who fought bravely and steadfastly to establish the republic that is in existence when the play's action takes place. When Brutus thinks of doing what is best for Rome and says in his funeral speech that he loves Rome more than Caesar, this is what he is referencing. Cassius knows that Brutus will do almost anything to maintain the republic's ideology rather than watch it slip away to Caesar's dictatorship or "kingdom."
Brutus is a noble, well-respected senator in Rome. Even though Brutus sided against Caesar when Caesar fought with Pompey, Caesar gave Brutus a seat in the Senate. Caesar did this for two reasons: he knew Brutus could be very dangerous, as he was a well-liked Senator and had the support of the plebians, and because Caesar wanted to keep Brutus close to him, as Brutus was a powerful man.
Cassius knows that Brutus is well liked, so if Brutus supports his plan, so will the plebians who support Brutus. Cassius also knows that Brutus is close to Caesar. Shakespeare portrays Caesar as trusting of Brutus, which Cassius hopes he can use to the conspirators advantage. While Caesar may have been somewhat trusting of Brutus, the real historical Caesar was probably still wary of him.
As was mentioned in the previous posts, Brutus is a well-respected, popular Roman, who carries a significant influence with him throughout Rome. Brutus also comes from a prestigious family that helped establish the Republic. If Cassius and the others can convince Brutus to join their cause, they believe the masses will not rise against them. Since the majority of Romans identify with Brutus and support him because of his family's name, the conspirators feel that having Brutus participate in the assassination will be good for public relations. Also, Brutus's ancestors were the first to rid Rome of its initial tyrants. Having Brutus on their side would legitimize their actions in the eyes of the public. Eventually, Cassius succeeds in getting Brutus to join the conspirator's plot to assassinate Julius Caesar. However, Mark Antony gains favor in the eyes of the public following his funeral oration.