Shades of Rawls! This is not the thread to argue it, but one can make the case that behind the veil of ignorance we would choose the mean, not the minimum.
From behind the veil of ignorance we would choose the mean in utils, not the mean in dollars.
Now utils are a theoretical construct, and no one really knows how to convert dollars to utils, but I think we can all agree that the conversion is sublinear - your first dollar is worth more utils than your thousandth dollar which is worth more utils than your millionth.
It's also different for different people. My conversion scale is probably more sublinear than yours - the ratio between the worth of the thousandth dollar and the millionth dollar is bigger for me than for you. I do say that, if I ever become a billionaire, you should shoot me and put me out of my misery because I've clearly lost my mind.
Interpersonal utility comparisons are not unusual. We think we know, e.g., if one person is needier than another, i.e can benefit more from more money, benefit measure by his utility function. We put ourselves in others' shoes. Be that as it may, it is hard to imagine doing so systematically. Since utility is monotonic in income, no great harm is done by using income instead of utility and by assuming identical utility functions as a first pass.
Then the argument is that behind the veil we would choose institutions that maximize total
income because our chance of landing in any part of the income distribution is equal to landing anywhere else. The greater total income, the better off we are no matter where we land. This makes mean income the proper measure of social welfare. In other words, distribution doesn't matter. Pure utilitarianism.
Having done all this, we consider that we have established institutions to help the worst off. [It is perhaps surprising how generous these are even in the United States.] This redistribution makes society better off on account of the diminishing marginal utility of income. But of course this doesn't take us to equality of outcomes. Another way of putting this is that Rawls assumes too much risk aversion compared to real human beings.
So, I would be perfectly justified in using mean income to measure social welfare. I haven't. I used median income, as a tad more insurance against the risk of being too far in the bottom.