But maybe they just want to say 'we're angry about this rising income inequality and we want someone to do something about it.' I think that's a valid stance to have, and opinion to express. It then falls on the government to either:

1. Do something about income inequality. 2. Explain why they can't or shouldn't or won't.

Failing that either of these things occur, it may inspire someone to run for office with the platform of performing (1) or (2) above. Any of these outcomes seem like a 'successful' outcome of the protest.

Not necessarily; there are times when it's more fit to just walk out into the street, meet others who are as displeased as you are, and try to figure out what to do about it on the spot.

Another way to look at it is this: The grievances of the protesters cannot be fixed by individual action. They require a change of the rules, i.e. political action. When such grievances become large enough, you (i.e. the people) use progressively stronger methods to try to tell politicians that you're serious about it.

If the political system is flexible and effective enough to react and address the grievances, then the movement succeeds, e.g. the civil rights movement in the US in the 1960s.

If the political system is too rigid and unable or unwilling to react, and the grievances remain, then the state will ultimately fail, e.g. East Germany in the 1980s.