“The legacy of deportations is going to be with her for a long time to come, and it is something that she is going to have to live with,” said Antonia Hernandez, the president of the California Community Foundation and a former chairwoman of the Mexican American Legal Defense and Educational Fund, who spoke with Ms. Napolitano at the University of California, Los Angeles. “She is acknowledging that diversity is a weakness that the system needs to address. If it is only educating the wealthy upper class, there is not going to be a lot of will to put money into that kind of public institution.”

Complaints about deportations are just the beginning of Ms. Napolitano’s challenges. For years, the state has put less and less money into the university system, reducing the state’s contribution to levels not seen since the Great Depression — forcing the system to increasingly rely on tuition, government research grants and private dollars, even as it reduced the number of courses offered and cut back on library hours. In many ways, Ms. Napolitano’s primary job is convincing people who will never set foot on a University of California campus that the system deserves their tax dollars.

She is not the sort of magnetic leader who wins over skeptics with charm. But what she lacks in charisma she makes up in dogged efforts, inviting some of her harshest critics to intimate meetings and quickly deciding whether their criticisms warrant action. She travels with a beefy security detail provided by the university. At 56, she has never married and is a self-described wonkish workaholic, telling audiences that she takes home budgets to read in bed.

That hard-nosed demeanor could make her the perfect match for Gov. Jerry Brown, who has become deeply involved in the university over the past year and is arguably the most important figure in its future. The governor, who served on the search committee that selected Ms. Napolitano, has insisted that a $140 million increase proposed in the governor’s budget this year be tied to a tuition freeze and an improving graduation rate.

Even as she continues to court the public, Ms. Napolitano’s most pressing long-term task is to convince the governor and state legislators that the system both needs and deserves more money to regain its luster.

“We need to make it clear that when we ask for something we’re not just going with our hand out to be a supplicant, but we’re saying this is an investment and here are all the many ways we are going to give you a return on that investment,” Ms. Napolitano said in a recent interview. “The days of just getting money are over. We have to be a place — and we are — that the state can turn to help solve its problems.”