I've said before that I would prefer to replace the minimum wage with a guaranteed minimum income, but I've since thought of a justification for keeping the minimum wage.
In any economy, there are commodities - things which are easily available and the prices of which are easy to discover such as gold, oil or milk. One very important commodity is the hourly labour of a human who is healthy and competent at following simple instructions but otherwise unskilled. Even without a minimum wage law, there will be a natural price for such labour. It will fluctuate with time and space but will generally stay within a certain range. In some times and places it might actually go higher than the legislated minimum wage (which corresponds to a state of zero unemployment).
Having a legislated minimum wage has the effect of setting a scale factor (or a "gauge", for the physicists) of the economy as a whole - it's sort of like having an electrical circuit that isn't connected to anything and then connecting one part of it to ground. If the minimum wage is set too high then it will cause an inflationary pressure which will dissipate once everyone has a job again. If it's set too low then it will have a negligible effect anyway, since there would be very few people who would be unable to get a job for more than minimum wage. According to this theory, the minimum wage has nothing to do with making sure the poorest members of society are paid a respectable wage (which makes sense since a minimum wage is actually a regressive policy) - it's just an economic control factor.
Now, as an economic control factor, a minimum wage has a number of problems. One is that it takes a while for the inflation to eliminate unemployment after the market rate for labour goes down, so there's always some residual unemployment. Another is that people are not all equal - the labour of some people is just naturally below the basic labour rate (because they are unskilled and also either unhealthy or incompetent). While this is unfortunate, essentially forbidding them from working at all (by preventing employers from hiring them at the market rate) seems to add insult to injury (not to mention creating yet another dreaded poverty trap). A third problem is that there are many other ways that governments "ground" economies - in the electrical circuit analogy they're connecting various bits of a circuit to voltage sources that correspond to "what we think this ought to be" rather than what it actually is, which seems like a good way to get short circuits.