What theory is suitable for a thesis dealing with the courage of white man (Atticus Finch) against prejudice to blacks in To Kill a Mockingbird?
If by "theory" you mean "literary theory" -- and I suspect that you do -- I think it won't be too easy to find one that matches your focus on the courage of one individual in the face of widespread prejudice.
The focus on individuals fully aware of their motivations and in full control of their actions runs against the grain of most literary theories (e.g. marxism and structuralism focus on the larger social structures, psychoanalysis focuses in part on our behaviors and thoughts that are outside our control, etc.). Similarly, the focus on doing the right thing runs against the grain of most literary theories. I would characterize your intial statement as leaning toward a humanistic, pre-critical approach to Harper Lee's novel To Kill a Mockingbird.
There are ways to modify your thesis and approach, of course. For example, you might apply a New Critical approach, focusing strictly on the world created within the novel and the language used to create that world (e.g. no references to Harper Lee's "intent" or "purpose" is allowed!). You could explore how Atticus seems to be the one person in position to fight the madness that afflicts nearly every other white person in the town and county. You may want to look at other posts on Lee's novel at enotes.com that discuss the parallels in the novel between racism and rabies. New Criticism usually works best with short poems, but it can be applied to novels. One strategy to consider using is to identify particularly rich passages and to perform close readings on them, almost as if they were themselves poems.
You may also consider applying a structuralist or archetypal approach that looks at how good and evil are represented in the novel. The novel is often praised for not conflating goodness with white skin, for example, but the novel does (I believe) trade in another equally problematic stereotype when it portrays the poor whites (e.g. the Old Sarum men) and the white "trash" (Atticus' and just about everyone elses' term in the novel, not mine, for the Ewells!) as the most racist and the most capable of inhumanity. The point of a structuralist or archetypal approach might be to point out the belief systems that are absent and the ones that are present in the novel.
The approach that I can most strongly recommend without knowing your interests in New Historicism. You may enjoy exploring how the novel may be read as a response to famous rape trials and/or lynchings in the 1930s and the 1950s.
If I haven't answered this question well enough, please feel free to message me. I'll definitely help as much as I can.
That would depend on what type of essay you are writing, so it's difficult to say.
If you are writing a literary analysis, your thesis could say something about how people must stand up for what they believe is right, even if this means they stand alone. Then you could focus your essay on the setting, characterization and conflicts.
If you are writing a persuasive essay, you could basically use the theme from paragraph above and then cite historical examples such as MLK, Ghandi and Rosa Parks.