In my setting, there is a small, rocky ring system surrounding an earth-like habitable planet. This ring was formed relatively recently (approx. 50 million years ago) from a small moon with a Moon-like composition hitting the planet's Roche limit. Given these conditions, could hotspots of certain ores and minerals form or be present in this ring system? Imagine a small band of rocks containing an unusually high amount of, for example, iron or bauxite, ripe for a mining vessel to set up camp for a few days and gobble up, or perfect for a band of pirates to hunt passing mining vessels. Such a hotspot would make a great location both plot-wise for interesting things to happen, and environment-wise for providing the classic sci-fi "dense asteroid field," but I don't want to include it in my setting if it's too glaringly implausible.
Problem number 1 is that you want the rings around an Earth-like planet. Such rings are likely to be comparatively small and short lived compared to Saturn's rings, for example... Earth just doesn't have a big enough gravity well for really impressing decorations. A big ring structure is also problematic for "habitable", as a continuous rain of debris from space may look pretty but it plays havoc with your atmosphere and ecosystem. Might be OK far from the equator, of course.
The other problem is that you need something that's really quite poorly held together... a big ball of snow-and-gravel, for example, and those sorts of things are less likely to be rich in interesting minerals. A nice mineral-rich moon is likely to be fairly solid, and a metallic one even more so.
That's not to say that it couldn't happen, of course. Maybe a mineral-rich or metallic moonlet was broken up by a collision with another body, or maybe a big asteroid had a similar incident but then re-accreted over time before being captured (though capture is less likely when the planet is small, and captured objects often end up in distant orbits well away from their Roche limits...)
The materials are likely to be evenly distributed throughout the ring, though you are likely to find occasional large fragments that would be much more convenient to feed into a smelter, as opposed to continuously chasing little bits of gravel.
I'd handwave it in, and not go into too much detail about its origins. You'll be fine.
Already quite a good answer from Prime, but to add to it.
There is no necessity to handwave, the body is still in process of destruction, maybe it was hit as Prime suggests so there are not one but few parts of it, let's say what was the core of that body, which by itself can be a core of something else.
Rings are formed by dust which was on top of it which is collected, or anything, before it was captured etc.
So all that, it is not even a handwavium, but just a possibility, rest is pretty much as per Prime answer.
Also, as a note, rich bands give you nothing, really. It is not like you can stop, take a funnel and catch it all, if the ships you are using do depend on orbital mechanics and have limits of delta-v, which for a cargo ship aka miner will always be a limiting factor aka problem, no matter how big the number is.
For different bands, a spot - it just not possible, as each band rotates with a different period and by that mixing stuff up(kinda), not even talking about other processes happening in the ring.
Moving from one point a band to another point of it, it's really a time-consuming process, I recomend KSP in that regard. Intercepting on trajectories from that planet to other potential places/planets is an option - not an easy one or super convenient but it is possible, especially empty ship catching up with loaded one - if done right guys have no chances if they do not dump cargo.
But if there are sweet spots, chunks - hide in them if you can't capture them permanently and have slave miners work for you there. Or better scam their money in a spaceport - clean and easy.