We never actually got to the critical question of "what is a strong wheel?" So that becomes today's mandate. First, let's look at some dictionary definitions of "strong."
Having the power to move heavy weights or perform other physically demanding tasks. Since wheels move us, and since the lightest among us weighs about 35 times what even a reasonably stout wheel set weighs, I'd buy into this. Rolling over obstacles like potholes and rocks, and dropping off of curbs and such could be categorized as demanding tasks, so this is relevant.
Possessing skills and qualities that create a likelihood of success. Sure.
Powerfully affecting the mind, senses, or emotions. This is the PR department's job, not the wheel's. We'll strike this one as not relevant.
Able to withstand great force or pressure. Well this seems like an arrow to the heart of the matter, doesn't it?
You get the point - there are different ways we can define "strong" and some of them are more relevant than others, but the last one is really what I think we're thinking of when we talk about "strong wheels." Please note, this is a safe place where the descriptor "bomb proof" is not allowed. Seriously. That is an inappropriate phrase to describe wheels.
Plenty strong
So what makes a wheel able to withstand great force or pressure, and have the power to move heavy weights or perform other physically demanding tasks? And what does success in those dimensions look like?
The easiest definition of success there is that the wheels simply not break under reasonable application of heavy weights, physically demanding tasks, and great force or pressure. A more demanding definition would be that they stay stable and true under reasonable applications. A highly demanding definition would be that they do these things within high performance parameters - not understeering, not exposing the components to cycling fatigue, not needing constant maintenance, and not banging into the brake pads or rubbing against the frame.
So let's start with the hubs. Hub shells should be forged and machined. Forging means you heat and then bang the crap out of the metal into the rough shape, which makes that metal stronger. Then you cut away the excess and create the shape that you want for the hub shell. All of the hubs we'd consider using are forged and machined.
That's one big hub!
The shape of the hub shell, which dictates the things we measured in our comparison chart, also affects wheel strength. Flange size, bearing placement and size, accuracy of bearing bore machining - these are all hub shell construction issues. Good hubs like the ones we use are all good in these dimensions, and many cheap hubs are notoriously sloppy in these dimensions.
The materials themselves are important. Different companies use different hub shell alloys. In some cases you have valid arguments for one or the other, and these are what you might call principled differences. In cheap hubs, you might see the cheapest available alloy in there. Box store beach cruiser hubs don't use the same alloys that I9 uses in their hubs. Then you have axle materials, and cassette body materials. One knock against DT hubs is that their cassette bodies are softer compared to some. I9 and Chris King have quite hard alloy cassette bodies, and White Industries just uses ti for theirs. Many people poo-poo steel cassette bodies because, come on, hubs that use them are "boat anchors!" But steel cassette bodies can be very effective...
For his weight, Joe is VERY strong
Then we move to rims. As we wrote a bit ago, we're taking a pass on the AForce C25 rims because they're too light. Properly, it's not that they're too light, it's that they have a defined strength to weight ratio, and putting their weight where it is leaves the strength part of the equation too low. So "weak" rims will exhibit a few behaviors - they compress a lot under tire compression loads, they exhibit instability for a given spoke count, and they frequently have problems with spokes pulling through them. Some rims have had a history of not being able to withstand tire inflation loads, and tires blow out on them. Deeper rims will enable stronger builds spoke for spoke because deep rims drive improved bracing angles, and because deep rims withstand tire compression better, and because the distance between spokes gets shorter for any given spoke count with deep rims, and because shorter spokes act stiffer than longer spokes all else being equal. If you want to use a rim with an ERD (effective rim diameter) of 600-ish, like a Stan's Arch, we're going to approach the rest of that build much much differently than if you are using a rim with a 571 ERD (RCG) or 551 ERD (Cafe Racer).
Rim material also plays a role. Carbon is anisotropic which simply means that its strength can be oriented in different directions. Aluminum alloys are isotropic, which means that they're the same strength no matter which direction you push or pull them. This ability to engineer strength along different axises means that carbon can be made stronger gram for gram and dimension for dimension. You can engineer a carbon rim to have massive hoop strength, which will make it resistant to compressive loads, without adding much shape (depth) or weight. With alloy, you need to either add weight to a given shape (which works inefficiently) or make a deeper section (which is more efficient) to make hoop strength gains.
Rim details make a difference. A lot of work is going on now into rim lip shape and dimensions. HED's Eroica rims have the "Fat Lip" feature, and carbon rims are adding more material to resist the physically demanding task of smacking into the ground.
And then we get to spokes. In some ways, spokes works like "you need x grams worth of spokes to build this wheel properly." There are ways in which fewer thick gauge spokes do the work of more thin gauge spokes, and this is what a lot of OEM wheel sets work toward - they use fatty big spokes so that they can have fewer of them and still look rad on the ship floor. But fewer thick spokes leaves you with fewer points of control over the build. The closer a spoke is to its neighbor, the more those spokes can work together to control the build. And the fewer spokes you have, the more important each one becomes. A crappy build with a couple of undertensioned spokes in a 21 hole rear wheel could leave you calling a cab with an unrideable wheel with broken spokes.
You can tune a build quite a lot with different spoke types and different spoke counts to ensure that the wheel is efficient, in that it's as light as it can be to perform its role in the expected use. To go much beyond that in explanation would make this way more long and boring than it already is.
And lastly, a lot of Internet forum all stars like to say that you should be able to constantly "abuse" a wheel forever and have it stand up. Two quick definitions of "abuse":
Verb - Use (something) to bad effect or for a bad purpose; misuse
Noun - The improper use of something
I don't agree that you should be able to abuse a wheel. A good wheel will be able to stand up to the expected use, with significant margin for occasional instances outside of the expected use. "Hard use" implies asking your wheels to regularly withstand great force or pressure. Abuse implies "hold my beer." They're very different things. Don't abuse your stuff.