Gov. Neil Abercrombie has brought in more than $2.4 million for his re-election campaign next year, building a formidable bankroll while other Democrats consider whether he is vulnerable in a primary challenge.

History suggests Abercrombie should be favored to win a second four-year term, but with his job approval rating under 50 percent, with discontent among environmental, progressive and labor interests, and with the state’s political culture unsettled by the death of U.S. Sen. Daniel Inouye, a contested primary is a possibility.

U.S. Rep. Colleen Hanabusa, whom Abercrombie passed over as Inouye’s successor, had been mentioned as a potential primary opponent by Democrats disappointed with the governor’s performance even before Inouye died in December.

Just one governor since statehood failed to win a second term — William Quinn, a Republican, in 1962 — and primary challenges typically end badly for the challenger.

Republicans have no obvious candidate on their bench not tarnished by a lopsided election loss, so Abercrombie’s biggest threat may come in a primary.

Abercrombie, who raised more than $4.5 million for his successful campaign in 2010, will have the early fundraising advantage and the benefit of incumbency. The governor raised $2.4 million through the end of last year — $297,300 in the past six months — and had $1.4 million in cash on hand, according to a state campaign finance report filed Thursday.

In the first two years of his term, Abercrombie presided over a state budget that went from deficit to surplus, largely due to Hawaii’s economic recovery. He has argued that only now, in his third year, does he have some financial ability to act on "New Day" initiatives, such as early childhood education, greater energy independence and repairing the state’s aging infrastructure.

"The theme will evolve, but it will be a continuation of our ‘New Day’ initiatives," William Kaneko, Abercrombie’s campaign manager, said in an email. "We will focus on the governor’s programs that have already resulted in an improved economic outlook; the stabilization of the state’s fiscal budget; and progress in education, energy independence and health care transformation."

Kaneko said there is no drawback to robust competition in a primary, but asked about a possible Hanabusa challenge, he pledged to support her re-election to Congress next year.

"Our efforts are focused on fundraising and strengthening the Democratic Party’s base to recruit and support a new generation of political leaders," he said. "As the titular head of the Democratic Party of Hawaii, the governor believes it is critical to keep the congressional delegation united behind President Obama’s vision for our country. Rep. Hanabusa has built seniority in Congress, and is taking leadership roles on issues important to Hawaii. We will support her re-election campaign to Congress in 2014."

Inouye, just before he died, asked Abercrombie in a letter to appoint Hanabusa as his successor. The governor considered Inouye’s request but instead chose Brian Schatz, his lieutenant governor.

A source close to Hanabusa, speaking on background, said that some Democrats who are unhappy with Abercrombie’s leadership had been talking to Hanabusa about running for governor in the months before Inouye died. Labor interests, in particular, the source said, have encouraged her to run.

Hanabusa has not ruled out primary campaigns for governor or for the Senate against Schatz. She will likely make a decision by summer because of the fundraising demands of either path.

"Right now I am considering my options and keeping an open mind," Hanabusa said in an email. "I’m not ruling anything out, but I’ve had a lot of people expressing opinions and offering insights, so there’s a lot to think about."

Political analysts say Hanabusa would have to overcome the perception she is running against Abercrombie — or Schatz — out of pique at being passed over for the Senate appointment. While Hanabusa would likely find traction with organized labor, she would have to move significantly to the left politically to outflank Abercrombie among environmentalists and progressives despite their disappointment with the liberal governor.

"It takes a lot more than anger to challenge a sitting governor," said Neal Milner, a University of Hawaii-Manoa professor emeritus of political science. "It takes some kind of issues that differentiate you from the other person, and it takes a real weakness on the part of the incumbent. And, of course, it takes money. And as it stands right now, those things seem to be pretty formidable challenges to her, assuming that she’s even interested."

Tony Gill, chairman of Oahu Democrats, said primary challenges are tough.

Gill’s late father, Lt. Gov. Tom Gill, was a progressive who unsuccessfully ran against Gov. John Burns in 1970, a cautionary example for many Democrats even today. But Gill said Inouye’s death has left the majority party’s competing tribes without a center to maintain order.

"I think it’s become clear to people in the aftermath of the late senator’s demise that, lo and behold, the party has been transforming itself into a different kind of entity for the past five or 10 years," he said. "And I think that the death of Sen. Inouye allows some people to see that more clearly — the formal center of gravity is missing."