I'll give it a shot. Note that I'm going to discuss wealth inequality, not income inequality. (Because the discussion is almost always really about wealth, and not income.)
The con side: Wealth inequality lead to resentment and multi-tiered systems; the rich get better healthcare, for example, and therefore live longer. It is supposed that it leads to hardening class lines, as well; if only the rich can go to Harvard, and only Harvard graduates get rich (a gross simplification, but you see the basic idea), then class mobility goes to zero, which leads to declining meritocracy in society, which leads to suboptimal economic organization.
The pro side: Wealth inequality is meritocracy in action; it represents the tendency of those who are good at managing money to acquire more money (to be managed), which represents optimized (although not necessarily optimal) economic organization. It is, by dint of lost ability, a greater societal tragedy when an expert in managing wealth dies than an average given individual; therefore it's not necessarily a bad thing that wealthy people get better healthcare, given that healthcare is a finite resource. It is supposed that ability in managing wealth is necessary merely to maintain it, and therefore class mobility is not as rigid as opponents of wealth inequality would argue.
Which leads to a second con side, ability in managing wealth as the sole social-value determinant is suboptimal; Richard Feynman contributes more to society than your average hedge fund manager, the argument might go, and his investment of wealth, while not generating more wealth for him personally to invest, would generate more wealth overall.
Which of course leads to the second pro side, that ability in managing wealth is the only inherent property identifiable in our current system. That the system isn't perfectly optimized is a nature of imperfect information; we wouldn't necessarily recognize a new Richard Feynman, and even if a new Richard Feynman could use the wealth more wisely, the number of non-Richard Feynmans incorrectly identified as a new Richard Feynman would result in negative utility compared to the current system.
The argument can go on for ages. This is without discussing the influence of wealth in politics, which has a tendency to get heavily mired in color politics, as both sides think the other is unduly influenced by wealth.
It is supposed that ability in managing wealth is necessary merely to maintain it, and therefore class mobility is not as rigid as opponents of wealth inequality would argue.
How many people born to very rich parents end up very poor because they are bad at managing wealth?
Seriously, when I look at wealthy people, I mostly see people born from the right vagina. Or in some case people with good genes but mediocre memes (think about sportspeople or models).
In line with the results of the poll here, a thread for discussing politics. Incidentally, folks, I think downvoting the option you disagree with in a poll is generally considered poor form.
1.) Top-level comments should introduce arguments; responses should be responses to those arguments.
2.) Upvote and downvote based on whether or not you find an argument convincing in the context in which it was raised. This means if it's a good argument against the argument it is responding to, not whether or not there's a good/obvious counterargument to it; if you have a good counterargument, raise it. If it's a convincing argument, and the counterargument is also convincing, upvote both. If both arguments are unconvincing, downvote both.
3.) A single argument per comment would be ideal; as MixedNuts points out here, it's otherwise hard to distinguish between one good and one bad argument, which makes the upvoting/downvoting difficult to evaluate.
4.) In general try to avoid color politics; try to discuss political issues, rather than political parties, wherever possible.
If anybody thinks the rules should be dropped here, now that we're no longer conducting a test - I already dropped the upvoting/downvoting limits I tried, unsuccessfully, to put in - let me know. The first rule is the only one I think is strictly necessary.
Debiasing attempt: If you haven't yet read Politics is the Mindkiller, you should.