Irony. I once heard this great definition of irony from a guy named Matt Bird on a podcast called The Narrative Breakdown. He defined irony as "any meaningful gap between expectation and reality." This can mean many, many things, in equally as many ways, but overall, I would say things that are "ironic" are basically "opposites" of each other, but tied together by someone's expectations.
This GAP between expectation and reality (or what happens) is the fuel for nearly all emotional reactions, and is one of the most powerful parts of great story telling. And your Main Character could benefit by having at least one greatly ironic trait. Perhaps their most dominant trait. For instance, a character that is so smart he could save the world, but also so smart that he doesn't fit in and is scorned by that very same people he could save.
Not only could your character's strengths and weaknesses be linked ironically, but YOUR CHARACTER'S MAIN GOAL COULD BE IRONIC. For instance, you could have a man who is trying to kill another man to avenge the murder of his father, only to find out that the "murderer" is his father. (I know, it's cliche, but that's because it's so damn good).
This concept of ironic traits in your characters is tied closely to their INNER CONFLICT and OUTER CONFLICT as well. For instance, you could have a character whose very life depends upon saving a princess from an endlessly tall tower, except he's deathly afraid of heights (because he doesn't want to fall and die).
In short, having the the basic traits of your main character be deeply ironic gives you some automatic CONFLICT to work with. And that's always a good place to start.