What is the role of free will in The Hobbit? I am having trouble finding examples.
I think any discussion of free will in The Hobbit should naturally focus on Bilbo.
Free will, as a narrative or literary device, is often used to show the strength of an individual's choices weighed against their "destiny" which may be a magical force, or a social force. For example, Bilbo's free will might be weighed against the social pressure he feels from other Hobbits NOT to be adventurous or outgoing, and he might also be wondering if a Hobbit is, by definition, not suited for adventuring, and if his actions are likely to result in injury or death for himself and others. He does spend a lot of time feeling sorry for himself and wishing he was back home, and yet he doesn't actually turn back, which might be considered a subtle reinforcement of his free will to see his decision through to the end.
I think a more visible act of free will is his choice to take the Arkenstone as his fourteenth share of the treasure. Many of the other characters, Thorin particularly, rarely act against their "nature" or their sense of duty. The elves and dwarves maintain a cautious distance from one another, everyone regards Gandalf with awe and some suspicion, and Smaug regards everything and everyone as inferior. Bilbo is the only character to regularly show compassion to those who wrong him, and constructive anger at those on his own side. This culminates in his actions with the Arkenstone; no dwarf would, presumably, consider themselves capable of this act of apparent betrayal; it is even apparent that the stone has almost supernatural control over the dwarves, awakening their worse nature.
So, in short, Bilbo often appears to be the only person truly capable of a full range of justified and objectively noble actions; as Gandalf puts it, "There is always more about you than anyone expects."