I think the best argument against a minimum wage law is that it uses the force of government to prevent an otherwise willing man performing a job for less than the minimum hourly wage.
Assume the minimum wage becomes $10.10. I want someone to clean up my yard but don’t want to pay more than 9 dollars per hour. You want to do the job for 9 dollars per hour, but the government tells you that it is illegal for you to take this job! It has become illegal in this case for you to work for pay. What right does anyone have to tell a man that he cannot work for another man for a wage mutually agreed upon? No right at all, obviously. Therefore, a minimum wage law violates the rights of workers.
It’s really worth pointing this out, even though it’s nothing but the other side of the coin that’s always pointed out by those whose hearts bleed for the poor and the downtrodden. Yes, you are restricting the ability of the bloodsucking capitalist to “exploit” labor, but that’s not all you are restricting. But then, the bleeding heart elitists usually know that the poor and the downtrodden don’t (can’t?) know what’s best for them – so “we” need to have a compliant and wise government step in on their behalf.
I’ve explained this side of the coin to the man on the street on a few occasions, and I usually get the response “I never thought about it that way.”