I would argue that the central idea or theme of Trevor Noah's Born a Crime is that any hardship can be overcome.

One of the first things we learn in the story is that thanks to the draconian laws of Apartheid in South Africa, the title of the book is literally true: Trevor Noah's birth was a crime, and for parts of his childhood, he had to stay hidden. White people (like Trevor's father) and Black people (like his mother) were not allowed to have sexual relations. However, he overcomes this hardship, and this did not deter him from having many childhood memories of playing with friends and cousins.

Trevor's mother, Patricia, is the epitome of resilience. She breaks all sorts of laws to have the life she wants, including having sexual relations with Trevor's father and later taking a job in the city and finding ways to stay in the city at night, despite this being illegal for Black people at the time. She overcomes many of the struggles of being Black in Apartheid South Africa.

Another hardship we see eventually overcome is Patricia's relationship with the abusive Abel. Although it takes him shooting her in the head after she tries to leave him, Patricia overcomes this hurdle, even joking with Trevor while in the hospital that he is now the best-looking member of the family.

We see the various entrepreneurial ideas that Trevor explores throughout his young years, such as burning CDs with his friends and working as a DJ. In this way, Trevor overcomes the fact that his family doesn't have much money.