Mitt Romney supports pegging the federal minimum wage to the Consumer Price Index. Yglesias figures that "[i]ndexing the minimum wage to inflation should appeal conservatives" and that it "would deny Democrats the political weapon of random increases." Reihan Salam revisits Ron Unz's comprehensive proposal for a substantial hike: 

I find it unimaginable that Romney would support a steep increase in the federal minimum wage. Yet were he to do so, he would devastate his Republican rivals. A significant minimum wage increase is backed by a large bipartisan majority, leaving aside its substantive merits. Unz emphasizes the impact on immigrant workers, though this policy would also have some nontrivial impact on teenagers, etc. 

James K. Galbraith recently endorsed raising the federal minimum wage to $12 an hour (the current rate is $7.25):

The plan isn't just good for Republicans — it's good for the economy.

What would workers do with the raise? They'd spend it, creating jobs for other workers. They'd pay down their mortgages and car loans, getting themselves out of debt. They'd pay more taxes — on sales and property, mostly — thereby relieving the fiscal crises of states and localities. More teachers, police, and firefighters would keep their jobs. 

Would this hurt competitiveness? Not at all. That's an issue for manufactured goods and traded services like insurance and banking, sectors in which everyone already earns far more than $12 an hour. The jobs we're talking about are in non-traded services like checkout clerks, haircutters, domestic help, and food-service workers — you can't run a deep fryer in Terre Haute from Bangalore.