To redouble something is to intensify it or make it bigger. After losing your initial run for school president, you might redouble your efforts the next year, with a bigger and better campaign.
You'll almost always find the verb redouble accompanied by efforts, as when the local police force makes a pledge to redouble their efforts to cut down on people texting while driving. Or in an attempt to have a more diverse student body, a college might redouble its efforts to recruit applicants from different geographical areas. The root of redouble is the Middle French word redoubler, which combines re, "again," and doubler, "to double."