The Republican mandate, if there is one, will vanish quickly. There are several issues where they could possibly get something passed, but each one comes with their own problems.
Immigration? Everyone wants something done, but no one wants to do anything. No matter what changes you make that Obama might sign, there will be someone who will label it as "amnesty". So if they move here, it will have to be a bill that they know the President won't sign. But in order for Obama to get the blame, they have to do it fast! Right now, their approval numbers are about equal. A Presidential veto of an unpopular bill could resurrect Obama's numbers which will hurt the Republicans.
Audit the Fed? We have to be realistic here. The only way this will ever come to the floor is if they are assured there are enough votes to stop it. (Or if it's watered down such that it is not a true audit) If they bring this up, it opens up anyone who votes against it for a primary challenge. And since they'll need at least some Republicans to vote against it, there will be pressure not to hurt their incumbents.
I can definitely see them moving on Keystone and I bet Obama signs it. This will give them both a small boost and will take the issue off the table. Unfortunately, having that issue on the table has helped the GOP. But if they get credit for resolving it, it could help them. There are other energy-related proposals that could do the same thing. I wouldn't expect too many of them to actually have economic impacts ahead of 2016, though, so very little credit will be given for improving things.
I would love for them to bring up Rand's REDEEM act and his Repatriation bill. Both of those things could actually get signed by Obama. However, there will be pushback from the establishment. The Tax Repatriation Bill could actually have immediate positive economic impacts. The problem is that if the economy is improving, the President will likely take credit. Bill Clinton employed this strategy after getting his ass handed to him in '94. The REDEEM Act would put many of Rand's own party in an uncomfortable position. They'll see it as slicing their own throats by restoring rights to ex-convicts. In some states, that will allow them to get their voting rights back. Even though it's a small number, the fear is that it could result in more democratic voters and other expanded benefits.
They could also try health care. Now, these are sure to be vetoed, but it will continue the fight into 2016. The fight has been helpful to Republicans, though, so they might push it.
Unfortunately, Obama is kind of in the driver's seat. He has options, too.
Push his executive over reach. This will cause combativeness. The GOP could get caught up in investigations and hearings. Then, the dems can run on the fact that the GOP was too busy playing partisan games than actually trying to get things done.
He could use his veto pen and the media to prevent any new legislation from being passed and painting himself as the savior from bad GOP policies.
He could go Bill Clinton and co-opt any economic changes the GOP passes. This could help his legacy and could help Hillary. I tend to think he'd like to do the former, but not the latter.
Anyway, should be an interesting couple of months to see which way the winds are blowing.