I have considered this on several occasions. What I arrived at
Some nice combo of price point and feature set.
The GBA and DS were genuinely good portable devices for the time, and they were pretty cheap and easy to come by.
Feature set wise.
My xbox connected to my TV at a time when a VGA port in my TV was a rarity and getting VGA out of a console was also reserved for those willing to go deep. It also had a PC like media player and emulators in a lot of cases. I am not entirely sure why they wii hit as hard as it did but it was pre raspberry pi, many had one, it was very easy to hack, came with a workable controller and was pretty cheap when all was said and done. The general sub par nature of mobile phone controls for emulation and gameplay styles seen on devices commonly emulated is probably the main driving force for a lot of 3ds work from where I sit.
Architecture and scene dev kit.
A bad architecture will hamstring you and while I enjoy messing around with assembly, reading hardware manuals and writing my own libraries there are others that want to press compile and have it go which leaves them to focus on getting things done, or maybe can not handle that low level stuff which is fine too as seeing everybody able to program like everybody can read today is a good goal for civilisation. The device maker having a kit leak out into the world, or the homebrew community offering something of similar or otherwise considerable potency is a key part of this. Having SDL helps too, the PSP having it definitely helped in several instances over the DS.
Having a similar architecture to something else of note may also help but more on that later.
Today we have if not entirely open then effectively open mobile phones, tablets, raspberry pis, crazy cheap PCs and TVs which have HDMI in as standard. If we ever have another proper homebrew scene (I do not count the 3ds as having one if I have to also include the wii, xbox, gba, ds and psp in the list) it will be because the device offers something unique and interesting. It could come from peripherals but I am not sure if anybody really has the juice to make a non standard device to support an interesting peripheral rather than just sticking it over USB or something, it could also be because the device has a unique ability that allows it to do something -- various arcade hardware and consoles offered significant advantages over contemporary PC hardware of the day (and for some time after in many cases) which allowed them to far exceed what said PC hardware could do, however I again deem this unlikely. Maybe if something appears that means we do not face the "up to 16 bit era it is great, after that it gets tricky" thing for emulation for however many years this is now but I doubt it.
I would not want to see it but there is a ghost of a chance some VR company pulls this off, I would be distraught to see good VR go locked down though.
I do also have to remember the effect the rise of ios (and then android) had on the DS homebrew scene, probably some of the PSP stuff as well but I was less aware of that. The injection of easier means of making money may not have been an overwhelming positive but its effects I can not ignore as part of this.
There will always be people that pull things apart to figure out how they work, pull things apart and tweak them to do something better for them, create software that does something that primarily interests them, create software that is not really commercially viable but still very cool and the other things we like and see in homebrew circles. A world where that does not happen is not one I care to be in*, and indeed is my main problem with a lot of science fiction/future fiction.
Whether we will be able to continue to view it through the lens of a specific console is a different matter entirely, and actually I am content to let it become part of history rather than trying to prop it up. Also though I mentioned raspberry pis several times during this a lot of what goes there is fairly basic ARM linux or even general Linux, and most things are ported out or ported to it. The closest to specific device stuff will probably remain like custom firmwares for phones and cameras is today, and that is hardly port your favourite emulator as much as hack something specific in.
*choice video
