I don't think this is motivated by a conversion to a Card-Krueger style belief that demand curves don't slope down (and nor perhaps should it be), or to the merits of wage-led growth. If this were so, there'd not be much left of conventional Tory economic thinking.
Instead, the justification for hiking the minimum wage could be the belief that the wage gains to the many more than offset the job losses suffered by a few.
To get a rough feel for how this is possible, let's use some estimates by Nattavudh Powdthavee*. He's calculated that a 1% rise in income raises well-being by 0.019 points, on a 1-7 scale. (I'm taking the fixed effects results in table 1).
If we take 100 minimum wage workers, you might think this means a 10% rise in the minimum wage would raise "utility" by 19 points - 10 x 0.019 x 100. You'd be wrong, because the higher minimum wage would trigger reductions in tax credits and other benefits. If we assume a withdrawal rate of 50%, we get a utility gain of 9.5 points.
Now, Professor Powdthavee also estimates that unemployment reduces well-being by 0.289 points. This implies that if fewer than 32 workers lose their jobs, then our 100 low-wage workers are, in aggregate, better off as a result.
This is plausible; it's unlikely that the price elasticity of demand for labour is as high as 3.2.
So, hiking the minimum wage is justified on utilitarian grounds, right?
Wrong. Higher wages for minimum wage-workers come out of employers' pockets, so they suffer a loss of utility. What's more, because higher minimum wages go party to the Treasury, their income loss isn't fully offset by workers' gains.
You might justify this by an empirical claim that the marginal utility of income is much lower for an employer than it is for an employee, or by a moral claim that low-wage workers' well-being matters more than that of employers. But I doubt a Tory would be comfortable with either.
Even if we allow for only a few job losses, therefore, it's likely that - from a Tory mindset - a minimum wage hike isn't a good utilitarian policy.
So why adopt it?
One possibility is that the public finances also enter Tories' utility function, so that a reduction in spending on tax credits becomes a good thing. This, though, is daft; in a liberal society, the state has no interests other than those of the people, so a transfer of income from employers to government can't raise utility.
The more rational possibility is that a rise in the minimum wage would steal Labour's clothes or shoot their fox - or whatever other vapid metaphor you prefer. If this is so, it means Tories are jeopardizing the (short-term) interests of some capitalists for their own advantage. Which raises interesting issues about the nature of the state in capitalist society.
* There are lots of ways you can quibble with this calculation, but I don't think doing so overturns my main points.