The Republicans in Senate will obviously start with the plan to run out the clock on Obama's term, because they'd rather gamble on having a Republican president in place next (which currently doesn't seem like a good bet with Trump in a pretty strong position for the nomination--his general election favorables are terrible, far worse than his Republican favorables, so he stands a good chance of losing to the Democratic nominee. Worse for the GOP--even if he wins, he's clearly not beholden to GOP dogma, so he's a wildcard to both sides).
However, there's a couple of issues with the Republicans simply blocking a nomination: one is that they only have the power to block in theory. In a vacuum, McConnell can simply not allow a vote. But nothing in politics happens in a vacuum; other Republican actions that they technically had the power to do--like shut the government down--collapsed when they realized how unpopular it was with the public. Like a government shutdown, a Supreme Court nomination is a big deal, not an obscure and arcane detail that most citizens aren't aware of. It'll be an interesting question whether the public agrees with the Republican position of expanding the so-called "Thurmond Rule" (which actually only applied to the last six months of a presidential term, not the entirety of his final year) and that they should block his nomination only 3/4 through his term, or whether this, also, proves to be deeply unpopular. The last thing the Republicans want is a wave election (of which there have been two recently--a Democratic one in 2008 and a Republican one in 2010--the 2008 one handed the Democrats just enough power to get Obamacare through, something you can bet Republicans haven't forgotten). As it is, the Republican majority in the Senate is not secure--there's a significant chance that the Democrats could win it back this year, though the odds are probably slightly below 50%. If blocking Obama's nomination all year is too unpopular, they'll abandon it to avoid tipping more Senate and House elections to the Democrats.
The other thing to consider is that blocking the nomination leaves the Supreme Court in a nominal 4-4 ideological tie (Kennedy is generally more likely to vote conservative, though he's not nearly as reliable as Thomas, Alito and Roberts). Ties in the Supreme Court leave the lower court rulings intact, with no precedent set. That places the power effectively in the Circuit Courts and, especially, the DC Court of Appeals--the most powerful court other than the Supreme Court. Over seven years of Obama judicial appointments, the DC Court of Appeals tilts significantly to the Democrats and the majority of the Circuit Courts are majority Democratic appointees.
The long and the short of it is that the Republicans don't reap as much as advantage as it may seem at first glance by blocking the nomination. They give up a year of mostly favorable judgments, with SCOTUS ties affirming major liberal victories in the DC Court of Appeals and Circuit Courts. If popular sentiment turns against them at all, Republicans may well decide the lesser harm is in allowing a moderate appointment, someone who doesn't completely tilt a Court that is currently mostly balanced. They'd never confirm a Kagan or Sotomayor under these circumstances, but Obama's expected Justice-in-waiting isn't actually as liberal as those two. It's been widely speculated for a few years that his preferred nominee in the case he gets to appoint another is Sri Srinivasan from the DC Court of Appeals. Don't let his foreign sounding name fool you into thinking he's a bleeding heart liberal! He's largely viewed as highly qualified and non-ideological. He was confirmed to the DC Court by a 97-0 vote...no one voted against him, not a single Republican.
