The minimum wage was originally created as a way to support a family. The minimum wage has not kept up with inflation and as such no longer can serve that purpose. Increasing the minimum wage would help decrease the amount of people needing government assistance which decreases government spending and increases taxes paid to the government. Happier people results in better workers.
Here's another point that came up with the decline of gas prices:
The more money the average american can spend, the better the economy. Our mainly-capitalist economy thrives when the average person can buy things.
If we're all in debt trying to pay for college, and unable to find a place to live off minimum wage, that's a huge dent. We need a fix. Raising the minimum wage is part of that fix.