The economics of minimum wage are pretty darn complicated. More than I understand, anyway.
The good ole' wikipedia has an introduction:
Minimum wage - Wikipedia, the free encyclopedia
The argument in favor of minimum wage that works best for me is the fact that it provides a good minimum floor for wages. Otherwise, low-skilled people have precious little bargaining power in employment situations. It's depressingly easy for employers to exploit employees otherwise. Minimum wage is at least 'some' money, even if it isn't enough to live on in many parts of the country.