What does Mark 10:50 mean?

A cloak was a wide vest that reached to the ankles and was worn as the outer garment. In the Bible, it's also called a robe or mantle. They were worn by everyone including priests (Exodus 28:31), kings (1 Samuel 23:4–6), and women (2 Samuel 13:18), although apparently there were distinctions between those worn by men and women (Deuteronomy 22:5).

Cloaks play a significant role in the Bible. The Mosaic law states that if a borrow gives a lender his cloak as surety against the loan, the lender must not keep it overnight, as it may be the only shelter the borrower has (Exodus 22:26–27). When Elijah is taken to heaven, Elisha takes Elijah's cloak as a symbol that he has been granted that prophetic authority (2 Kings 2:13). Isaiah prophesies that the kingdom of Judah will see such hardships that a man who still owns his cloak will be considered well-off enough to lead (Isaiah 3:6). When Paul, imprisoned in Rome, senses winter is coming, he asks Timothy to bring him his cloak (2 Timothy 4:13).

The fact that Bartimaeus leaves his cloak on the ground gives an indication of how much he trusts Jesus to heal him. The road is crowded, and he is blind. Even if he knows the area well from long days begging, Bartimaeus will be in dire straits if his cloak is kicked along the road or stolen. We don't know if Bartimaeus goes back for his cloak after he is healed or not. Now, he's not too concerned about it.