Mark 10.49-52:
And Jesus stopped and said, “Call him.” And they called the blind man, saying to him, “Take heart. Get up; he is calling you.” And throwing off his cloak, he sprang up and came to Jesus. And Jesus said to him, “What do you want me to do for you?” And the blind man said to him, “Rabbi, let me recover my sight.” And Jesus said to him, “Go your way; your faith has made you well.” And immediately he recovered his sight and followed him on the way. (ESV)
It is a reasonable inference to conclude that Bartimaeus had heard stories about Jesus. Stories of Jesus healing sick people in Galilee, and it appears that Bartimaeus had already concluded that Jesus is the Messiah. When Bartimaeus heard that the commotion in the crowd was because Jesus was passing by on the road, he did not deliberate as to what to do. He called out to Jesus in a way that indicates a knowledge of Jesus being a descendent of King David. It was revealed in Isaiah 11 that the Messiah would be a descendent of David, and the Messiah would thereby sit on David’s throne (Isaiah 9.6-7). Matthew 1.1 confirms that Jesus Christ is the son of David, and is thereby the fulfillment of these prophecies concerning the Messiah.
So Bartimaeus had concluded that Jesus is the person who has the authority to have mercy on him. And Bartimaeus expected that this mercy could and would be worked in a way of restoring his eyesight. In the Gospel accounts the working of miracles and healings by Jesus were associated with the faith of the person being healed. Such a person believed Jesus could do this. This faith was indicated in a variety of ways, but it seems to be evidenced frequently through the action of the people coming to Jesus. This would be their opportunity for healing and they took action to seek it from Jesus.
In the case of Bartimaeus, I think his faith was evidenced by his repeated calling out to Jesus for mercy. He would not be denied. All of the people telling him to be quiet could not back him down. And Jesus affirmed to Bartimaeus that he did in fact have the faith to receive this healing.