The blind beggar in Jericho (Mark 10:46-52) was engaged in something demeaning. His name is mentioned in the Bible as Bartimaeus, which means son of honor. Though "a son of honor" he was doing the most dishonorable profession of begging in the city of Jericho. When Jesus met him he was begging. His style of begging (like all beggars do) is to catch people's attention by crying loud "have mercy on me" so that they can throw a coin in to his bowl.
He did the same thing when Jesus passed by in the company of a large crowd. This time he did not just cry "have mercy on me" but was more specific because his plea was aimed at Jesus and not the crowd. He cries, "Jesus, Son of David, have mercy on me!"
However, it is the plea of a beggar. That is how the crowd who rebuked him to keep quiet took it. He should not nag Jesus for a coin! No beggar should bother Jesus, he has greater things to attend to! However, Jesus stopped, knowing that in that plea there is something more than a beggar's longing for a coin.
The crowd (who earlier tried to stop him) now encouraged him to walk towards Jesus saying "cheer up! On your feet! He is calling you." Probably, they might have thought that Jesus is going to give him a considerable amount.
The possibilities were limited: a donation to a beggar and nothing more. That's all that we can think at the moment. Even Bartimaeus cry was ambiguous. That's what makes Jesus ask him specifically: "what do you want me to do for you?" That is to ask, "Do you want a coin, another donation?" or something else. The beggar's answer was specific: "Rabbi, I want to see". Jesus can do both, he can toss a coin into the beggar's bowl. Any one who has coin to give away can do that. However, Jesus can also do something else; that is to deal with his problem at its root.
Jesus demands us always to make that choice. Do we want Jesus to deal with our immediate, felt needs alone, or work at the root of our problems. A coin or two that Jesus could toss in to his bowl may make him happy for a day. However, dealing with his blindness, the reason for being a beggar will change his stigma of a beggar and bring the honor back to him. Bartimaues made the wise choice and challenges us to make a wise choice in our lives too. Jesus poses the question to us because we are the ones who should make that choice.