Disproving a stereotype that was applied to oneself
Achieving a goal people said it would be impossible to accomplish
Changing someone's mind about a moral issue
Overcoming the lies one grew up with (that one was stupid, lazy, worthless, etc.)
Rising above one's circumstances (proving the culture or society wrong)

There are five basic human needs that, when missing from a character's life, could motivate them to pursue this goal. The following needs are all possibilities, but only one of them should be the primary driver for any given character. For more information on the relationship between human needs and outer motivations, please see this Character Motivation tutorial.

Self-Actualization: This need can be a motivator when the character is being held back. If the false beliefs keep them from pursuing their dreams or rising above their circumstances, they may set out to prove wrong the people perpetuating that idea.
Esteem and Recognition: Very often, when we're consistently told a lie, it's difficult to reject. When that lie has to do with who we are, it becomes part of our identity. A character who becomes tired of feeling badly about or being being ashamed of themselves might pursue this goal as a way of regaining their self-esteem and the respect of others.
Love and Belonging: If the character has become marginalized due to certain perception or beliefs, they might choose to prove those ideas false so they can fit in and find love.