"Social mobility is a measure of a just society." It needn't be. Imagine a dictator who imprisons his subjects, but gives wealth and power to some chosen at random. There'll be a lot of social mobility, but no justice or liberty. The test of a just society lies in the actual lives led by people - or by the worst-off if you're a Rawlsian - not by the chances they have. I'd rather live in a free, wealthy and moderately egalitarian society without social mobility than an unfree, poor and unequal one with it.
"Efficiency requires that we get the best people into top jobs." This is doubtful. For one thing, social mobility might worsen the quality of leadership. People who rise from humble backgrounds to "top jobs" might well become overconfident about their abilities and so prone to reckless decision-making. If your looking for poster boys for social mobility, you could do worse than Fred Goodwin, Hank Greenberg or Charles Prince. And for another thing, it could be that the problem with many organizations isn't that they are led by duffers but rather than they simply cannot be managed well from the top-down because they are inherently complex, or because hierarchies naturally demotivate employees. If so, it might be more efficient to restructure organizations so they become less dependent upon scarce "leadership" "talent."
"It's only fair that people get an equal start in life." This, though, is a utopian fantasy; you can't equalize the quality of parenting. And we shouldn't see life as a single race. Many people don't want to compete, but rather want to lead a modest life well. For these, the problem isn't a lack of social mobility so much as low wages, job polarization and the slow death of the goods of excellence. As Will says, rather than pursue a fictional equality of opportunity, we should have many spheres in which people can thrive by different standards.
In saying all this I don't mean that people should know their place. I'd rather have an egalitarian liberal society and let social mobility fall as it will. It should be a by-product of a good society, not an end it itself.
So, why does Cameron appear to think otherwise? Here's a conjecture. He wants social mobility for the same reasons that Michael Young - coiner of the word "meritocracy" - didn't. If we had equal opportunity, rich and poor would both feel that they deserved their place. In this sense, social mobility can be used to legitimate inequality.