How much of a majority?
Not a big one, and nowhere near the 2/3 necessary to override the veto power of the President.
Therefore anything meaningful would be compromise. Spending restraint, balanced budgets might be in the realm of the possible.
Supreme Court appointments would be interesting.
I believe the best chance for a Republican President is from among the Governors with track records. Walker, Perry come to mind. And an Hispanic VP could be a key.