Not a big fan of the Republicans, but it's fun to discuss how to make them electable. Oh, and my suggestions are strictly what I think will help them get elected, not what I would like them to do.
Things they could do to win votes (regarding the next presidential election).
1: Support immigration reform. Yes, it'll be controversial within the party, however, those Latinos are going to keep increasing their share of the vote, so you better find something to woo them. That and the business wing will love it.
2: The day Obamacare takes effect in earnest is the day getting rid of it is going to be nigh on impossible. Sure, reform it if you want, but remember why you where terrified of touching health care for the elderly, the same applies here to a large extent. Bonus: this will also help with the Latinos.
3: You are on the wrong side of the electorate when it comes to abortion. Do something about it but remain more restrictive than the democrats (to keep social conservatives from deserting). Bonus: women will like this.
4: Gay marriage is so far almost a tie, but it'll get worse, so either do something about it now or after the next election.
5: Leave drugs alone, you are on the side of the majority of the electorate for now (and among your voters especially so). Really wish it was different. Alternatively, do not prosecute those using drugs where said drugs are legal, while being officially against it.
6: Don't bother with the blacks, they are not increasing their share, are mainly located in safe states, and are going to be a nightmare to persuade.
Checking out seems not to do much.