What is the economic purpose of increasing the minimum wage? It is generally accepted among economists that minimum wage warps the equilibrium point between the supply and demand of labor by instituting a price floor and increases unemployment for unskilled laborers (while increasing the wage for the individuals who continue to work).
However, I have noticed that politicians continue to advocate an increase in the minimum wage.  Politics and political maneuvering aside, is there an economic reason as to why a higher minimum wage would benefit the economy?
Opinion of Economists

What makes you think that economists are so aligned against the minimum wage? Take a look at the IGM Forum that polls top academic economists. There is substantial disagreement about the effects and welfare implications of a minimum wage hike.


IGM Forum: Minimum Wage
NYTimes Blog Post about it


Theory about the minimum wage

Also, it's worth pointing out that besides empirical studies (like Card and Krueger), there are in fact theoretical reasons why a minimum wage might be welfare improving---see Lee and Saez (2012). (I'm just saying they exists, not commenting on the strength of the results.) In the Handbook of Public Economics, chapter 7, "Optimal Labor Income Taxation", they say this about that paper:


  Lee and Saez use the occupational model ... with endogeneous wages and prove two results. First, they show that a binding minimum wage is desirable
  under the strong assumption that unemployment induced by the minimum wage hits the
  lowest surplus workers first. ... 
  
  Second, when labor supply responses are along the extensive margin only, which is
  the empirically relevant case, the co-existence of a minimum wage with a positive tax rate on low-skilled work is always (second-best) Pareto inefficient.A Pareto improving policy
  consists of reducing the pre-tax minimum wage while keeping constant the post-tax
  minimum wage by increasing transfers to low-skilled workers, and financing this reform
  by increasing taxes on higher paid workers. Importantly, this result is true whether or not
  rationing induced by the minimum wage is efficient or not.This result can also rationalize
  policies adopted in many OECD countries in recent decades that have decreased the
  minimum wage while reducing the implicit tax on low skill work through a combination
  of reduced payroll taxes for low skill workers and in-work benefits of the EITC type for
  low skill workers.