Such headlines are always interesting discussion starters because they give the lie to things you might have heard about how anyone that understands basic economics thinks X. Even if you ultimately decide they are misunderstanding something, the reasons a Nobel laureate will take a different stance than you might are always going to be thoughtful.
If by "U.S. minimum wage" we mean federal minimum wage, I find it almost impossible these days to get enthusiastic about an increase. It doesn't keep me up at night worrying about it, but I really can't be super supportive of it. The empirical evidence clearly says that there are no real employment effects (or small positive effects if you find the methods of a certain set of studies persuasive, or small negative effects if you find the methods of another set persuasive). We are all entitled to prefer one method to another if we have our reasons, but the big picture is it's not a big effect.
You could argue that a nil effect argues in favor of increasing the minimum wage. I don't think that's entirely clear cut. There are other margin adjustments as well as long-term adjustments that we may be worried about. But even putting that aside, this is still going to be an average effect. Surely we think that a federal minimum wage imposed on a country of three hundred million people is going to be more binding in some areas than others. If you had a big positive effect maybe the average effect thing wouldn't bother you so much. But a nil effect means that the distribution of local effects is going to be both positive and negative.
And why wouldn't it be? You're saying that the price of low wage labor in Los Angeles, California should cost the same as in Oxford, Mississippi.
The nil effect in the empirical literature tells me there's something to the monopsony arguments. The data just don't make sense otherwise, and the theory is reasonable enough so I have no reason not to believe that. So if states want to raise their minimum wage, that's fine by me. There might be some free lunches to serve up, and it's their prerogative to go after them.
But I don't see why I should feel the same way about the federal minimum wage.