It is no secret that Democrats don't do a great job running the country. It is also no secret that Republicans don't do a great job running the country. At any given point, the only thing almost all Americans agree on is that the government is ineffectual. Democrats have too much to lose to take advantage of this moment in history, so I suspect they will stick to the status quo. The Republican Party, however, is not doing well, and one could make the argument that the party has nothing to lose.
Republicans, why not focus less on unsustainable initiatives, like anything that means fewer voters at the polls, and instead find issues to reach people on and run with them before the party falls into obscurity? Rand Paul seems to agree with that logic, so why can't the rest of the party get on board? The new voting laws being pushed in states across the country are not just about requiring ID. Many of the laws take aim at limiting or eliminating early voting (which has consistently increased voter turnout), as well as making it very hard for people to register voters during registration drives, which has also brought many voters to the polls in the last few years.
There are so many gaps in our country that need to be filled, but instead our government is tied up in its own dysfunction. I see this as an opportunity to reclaim the Republican tradition of progressivism. Progress does not mean Democrat, liberal, young, brown or any other "leftist" construct. A progressive person is a person who seeks to make progress, simple. Why would any political party make it a goal to be on the opposite side of progress for the country?
So what are a few (of the many) gaps Republicans can take the lead on?
Race
No one in America knows how to be honest about race. We are guilty of past, present, and institutionalized racism. The latter two do not have to continue, but they will as long as we deny the former. We have done horrific things around race, and in most cases we have not addressed what that means for the progress (or lack thereof) of our nation. Instead of acknowledging our mistakes, we suggest that everyone has equal footing now because our laws no longer actively oppress certain groups. If no one figures out a way to talk about and address race in America, the problem is only going to get worse as we experience more diversity.
Weed
Needs to be legalized. Not decriminalized, not okay in medical circumstances, legal. Keeping it illegal systematically strips people of their rights. Politicians laugh off memories of smoking pot (or worse), meanwhile people are in jail for the exact same thing, and their families and hopes of a prosperous future are damaged. It is ridiculous that something so ubiquitous ruins lives (disproportionally for certain populations) in our system.
War
Many of the problems in the Middle East are beyond our repair. We caused some of it, but we also have nothing to do with much of it. The problem is, we've created entire populations who hate us for reasons you would hate someone for, and when they retaliate we do what we usually do when it comes to claiming responsibility, we don't. We may not be able to stay completely out of military situations in the Middle East, but we could work to address the innocent people we have hurt in these conflicts and how to show remorse.
Education
An educated citizenry creates a more stable democracy. Unfortunately, the trend in America at the current moment is anti-intellectualism, which is perpetuated by politicians who often benefit from the ignorance of their constituents. It is an unfortunate development since that attitude is so far removed from the intellectual foundation of the country. We have a system that discourages critical thinking and encourages conformity. That is a particularly un-American development, and incidentally, it is also un-Republican.
I don't like the idea of a single party system in the United States, and I actually like the stubborn history of the Republican Party. To remain relevant, Republicans must reach people in America now, not the people who lived here in the past. The current brand of conservatism may have worked for people historically (or not), but that was then, we need to progress to now.