Sen. Lindsey Graham (R-S.C.) is up for reelection next year. (Cliff Owen/AP)

The challenges facing the Republican Party as it heads into the elections of 2014 and 2016 were on stark display here this weekend as South Carolina Republicans gathered for their annual convention, an event that revealed a party in the throes of some internal strife.

The source of the argument seems to boil down to what it means to be a true conservative in the modern Republican Party, and whether the party needs to change in order to broaden its appeal, particularly to Latino voters.

Disagreement boiled over most publicly when a few hundred delegates attempted to seat a convention chairman who would help them remove Sen. Lindsey O. Graham (S.C.) from the party. Graham, who has earned a place in Washington as one of the key players in the party establishment and a frequent critic of the Obama administration’s foreign policy, faced resistance from conservatives upset with his work to overhaul the nation’s immigration laws and his support for President Obama’s Supreme Court nominees.

Graham is up for reelection next year, but many of his critics hope a more genuinely conservative candidate will emerge to challenge him in a primary.

The senator acknowledged his critics when he spoke onstage, but warned them: “We’re going to end up in the same boat, whether you like it or not, because there’s holes in the other boat.”

When he turned to immigration, Graham framed the forthcoming debate as critical to ensuring the nation’s security — and the future of the GOP. Citing the immigration status of the suspects in the Boston Marathon bombing, he said the Obama administration had poorly enforced current laws and failed to notice that one of the suspect’s friends had overstayed a student visa.

The crowd appeared to be in step with Graham until he began discussing what to do about the legal status of more than 11 million illegal immigrants. Almost immediately, several in the crowd shouted: “No amnesty!”

“No amnesty, I’ve got it,” he said, cutting them off. “Now here’s the question: After 2012, the question for this party is: Do you believe that we can grow the vote among all sectors of this country?”

Graham’s message was clear: Reforming the nation’s immigration laws is critical to increasing GOP support among the fast-growing Hispanic population, which generally espouses the socially conservative views held by Republicans anyway and could very well eventually support the party’s candidates.

“Conservatism is an asset, not a liability,” Graham said. “You don’t have to be embarrassed about your social conservatism. You just have to talk about it in ways that we can grow this party.”

Finding a way to expand the party’s appeal was clearly on the minds of delegates Saturday, with many conflicted over who should lead the party into the future.

“They have to be strong, honest, able to stand up for what you believe in and not be intimidated,” said Stephanie McKay, a legal aide from Bishopville, S.C. “Without someone like that, nothing’s going to change.”

Dorian Bucholz, a first-time delegate from Charleston, S.C., said determining the party’s next presidential nominee will be difficult “because there aren’t really faces sticking out being strong conservatives. But we’ve got to get our act together because 2016 is going to come fast.”

Almost to a person, the delegates agreed that they want new blood.

“Our party’s ‘it’s your turn’ attitude is over for presidential candidates,” said Walter Whetsell, a GOP political consultant who has worked for several presidential candidates. “It’s never paid dividends for us. These young, fresh faces have a legitimate shot at winning over these people.”

One of those newer faces is freshman Sen. Ted Cruz (Tex.), who fueled speculation about his own ambitions by speaking Friday at the state party’s annual Silver Elephant Dinner. Pacing the stage like a televangelist, he earned loud applause by discounting his Harvard Law education, joking about the missteps of Vice President Biden and blasting Obama for his health-care law and the Justice Department’s botched Fast and Furious gun-running operation.

Clearly wooing the home-state crowd, Cruz recounted the stories of William Travis and James Bonham, two South Carolina-born officers who led the ill-fated Battle of the Alamo.

“Thank you for the support South Carolina has given, then and now, as we fight side by side,” he said.

By Saturday morning, however, several people said Cruz should wait.

“He’s only been in office about five months,” said Randall Wallace of Myrtle Beach. “I feel like you have to have some experience before you go after the top job.”

Jumping into the 2016 race would mean Cruz is “trying to replicate Barack Obama,” said Peggy Geraghty of Hilton Head Island, S.C. “I never agreed with Obama going for president so soon.”

Geraghty said she’s more interested in Sen. Rand Paul (Ky.), but worries he might not have broad appeal. As for Sen. Marco Rubio (Fla.), another name often mentioned for president, she said she’ll reserve judgment, but added: “He does have sparkle, so if you have to have the sparkle, I suppose he’d be good.”

Still reminiscing about last year’s GOP losses, Samantha Rogers of Bishopville suggested that Rep. Paul Ryan (Wis.), a former vice-presidential contender, should give it a go: “I actually enjoyed him more than Romney,” she said. “I like the values that he pushed.”

Even Graham acknowledged the party’s shifting demographics in his speech, referring to himself as “a little, short, white guy” in a crowd of rising stars, including the state’s 41-year-old governor, Nikki Haley, an Indian American, and Sen. Tim Scott (S.C.), 47, one of two African Americans in the U.S. Senate.

As Graham concluded, he made another bid to satisfy his critics: “The only thing I’m good at is being me.”

“I don’t believe enough in myself to think that I’m always right and that anyone who disagrees with me is always wrong,” he added. “But I do believe, and I owe it to you, my party and my country to be honest: We need to regroup, rethink and come out anew with a conservative message that will allow this party to grow.”

Most of the crowd rose to its feet and cheered, but about 100 people sat quietly in a dark, back corner of the room, still unimpressed by what they had heard.

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