In my view, I tend to think that Ron Paul's presidency would be viewed as an embarrassment for the Republican Party, especially if he became POTUS during 2008 or 2012.
2012 is the most recent election and his last attempt for the office, so I'd like to think that if Romney's campaign imploded during the Primaries that conceivably the former Congressman could win the nomination if Gingrich and Santorum bow out as they did OTL. Barring a major scandal tarring President Obama, I don't see him able to win the votes outside of the Deep South. But for the sake of the thread, let's presume Benghazi erupts sooner and the GOP runs with it tarring Obama's reputation enough for Paul to squeak by.
I see President Paul trying to push for the abolishment of the Department of Homeland Security which would generate a lot of pushback from prominent Republicans and Democrats (specifically John McCain and Hillary Clinton). Championing the abolishment of the welfare state would probably earn him considerable ire from the liberal wing of the Republican Party. Likewise, pulling out of the Middle East would anger many in the party and even earn scorn from Democrats, though I suspect he would also have backing for this from peace minded people like Gary Johnson and Dennis Kucinich.
I'd be willing to bet that social conservatives would champion his pro-life stance as well as his anti-gun control stance (shoot even Bernie Sanders would support President Paul on that regard), but the real question is how many would support his going after the Federal Reserve? I wonder how many feathers that would ruffle. Some champion an audit of the Fed, but not many seem to argue for outright abolishing the Fed. You could see many talking about how Paul wants to damage the US financially by destroying the Fed, and some would likely point out the economic downturn that happened when Andrew Jackson abolished the Second National Bank.
You’d have a political firestorm though from his stance on the Civil Rights Act from people like Al Sharpton and Jesse Jackson which would tar the Paul Administration as inherently racist (think “George Bush doesn’t care about black people” on steroids.) In 2014, the midterms would not be kind to the President as the Democrats would likely maintain control of the House and gain control of the Senate which would see political obstruction that would make the Tea Party’s current filibustering seem tame. Meanwhile, President Paul’s approval ratings would likely be in the low 20’s nationally, enough to make people seriously question whether or not they would have been better off with President Obama or even a potential President Romney.
If we go by the positives, you might see President Paul champion something like term limits for the Congress and the Senate, but that would only really happen if he had Gary Johnson as his running mate, which is extremely likely. You might even see the legalization of cannabis, or at the very least decriminalization of it. But, for every positive, President Paul would have a lot more negatives to go with it, most likely the alienation of our allies in NATO as well as trying to pull the United States out of the United Nations. Though, the latter would most likely be championed by a handful of the conservatives in both parties.