With President-elect Obama's major victory and the Democratic gains in House and Senate, we will likely see a shake-up in the Republican party as they work to appeal to a larger portion of the electorate.
Here's my prediction- the Republican party will swing back towards the causes championed by the fiscal conservatives rather than the social conservatives, thus drawing back in a portion of the Libertarian vote (Ron Paul will have a hand in this, or at least be a target for assimilation by the Republican leadership even if he doesn't take part). A portion of the religious right will be so put out by being sidelined that they'll either form their own party or join another one as appropriate. Some of the evangelical leaders will even as independents, though this will be about as effective as the Green Party has been on a national level.
Two years from now: Democrats will retain a majority in house and senate, losses will be minimal if they exist at all. Republicans may make gains on the state level in several places across the country. Evangelical split will be beginning- at least two high profile races tracked by the networks will include one of the religious right candidates in addition to a Democrat and a Republican.
Four years from now: The Social/Fiscal conservative split will be well under way - Palin will be with the socials, of course. She will have a "party position", though likely not the candidate for president even though she'll run for it. Some other figure from the religious right will win their support and get approximately 5% of the popular vote in the general election. This will be enough to give Obama a second term as it eats into the base for the Republican candidate. Democrats will lose a few seats in the senate and house but retain a majority.
Eight years from now: The social and fiscal conservatives have tried to reconcile and made some limited success. Still the bruises and scabs from the hateful rhetoric of the past six to eight years will be difficult to fully heal from and the Democratic candidate will have an advantage going into the presidential election. Still, the pendulum will have started swinging back and it will be a tough fight. Some segment of the social conservatives will have splintered off and will be (pardon my French) absolutely bat-guana crazy. We'll all have a good laugh, though be scared at the same time.
I wouldn't bank money on this, and I haven't been around as long as other folks on this list probably have, certainly, but I think it's at least a reasonable starting prediction. Feel free to adapt and update based on the output from your crystal ball and tea leaf readings.