7 And among the rebels in prison, who had committed murder in the insurrection, there was a man called Barabbas. 8 And the crowd came up and began to ask Pilate to do as he usually did for them. 9 And he answered them, saying, “Do you want me to release for you the King of the Jews?” 10 For he perceived that pit was out of envy that the chief priests had delivered him up. 11 But the chief priests stirred up the crowd to have him release for them Barabbas instead. 12 And Pilate again said to them, “Then what shall I do with the man you call the King of the Jews?” 13 And they cried out again, “Crucify him.” 14 And Pilate said to them, “Why, what evil has he done?” But they shouted all the more, “Crucify him.” 15 So Pilate, wishing to satisfy the crowd, released for them Barabbas, and having scourged Jesus, he delivered him to be crucified.
As Mark's version of this event emphasizes, the crowd was driven to cry for Barabbas by the chief priests who were at the root of the push to have Jesus killed. Why would they choose Barabbas as their alternative to prevent Jesus' release? The biggest factor, of course, is that Jesus was a problem for them - he spoke against them repeatedly, criticized their teaching and their ways, and taught many things that contradicted what they taught.
But on the specifics of why the chief priests would choose Barabbas, I think there's a pretty easy answer to reason out, even if the Bible doesn't explicitly detail it. Barabbas, as John mentions, was a revolutionary. That meant he was trouble, but not for the chief priests. He was trouble for the Romans. He wanted to resist Roman occupation and authority in the name of Jewish freedom. It's not hard to get an oppressed people to cheer for someone who wants to end their oppression. This would have been a win-win in the eyes of the chief priests, because they were getting rid of a man who threatened to compromise their authority in exchange for a man who threatened to compromise the authority of their oppressors.