This remains a fiercely Democratic state, and Ms. Lingle, as she acknowledged in an interview, will have to overcome the considerable obstacle of having Mr. Obama, who was born in Honolulu, at the top of the ticket. And for all the problems facing Democrats and the governor, who has struggled with budgetary and personnel issues, there are signs that things might be looking up here: tourism appears to be on the upswing, and Mr. Abercrombie said he had gotten through a difficult first year and begun to turn the state around.

“We’re not going to send somebody to the United States Senate whose sole purpose is to undermine the president of the United States, particularly if the majority in the Senate is so slim,” Mr. Abercrombie said in an interview. “I think it will be a tough race because the money will come in on the Republican side. And of course, being Democrats, we have a primary contest. At the end of the contest, we’ll go as we always have gone here in Hawaii.”

Still, the Republicans have in Ms. Lingle what party officials on both sides say may be the one Republican who could be elected senator from Hawaii. She is well known and is an experienced campaigner with access to huge sums of money. The Democrats are now split by an ideologically caustic primary battle between Mr. Case, who presents himself as a moderate in a party that has swung too far to the left, and Representative Mazie K. Hirono. Mr. Abercrombie’s continued political troubles have worsened the atmosphere for Democrats.

“This is the first time since 1977 when we can potentially have a Republican senator,” said David Chang, the state Republican chairman.