dtgreene: * The games on them are legal. This is important to some people; not everyone is willing to pirate games, and some of those who do pirate games and like them may want to go the legal route. (In my case, I decided that, due to the lack of an internet connection on these devices, they qualify as not having DRM.)
Legal shmegal. The target audience, including myself, has likely already purchased most of these classic games multiple times, across various platforms and versions. That's especially true for Nintendo's retro consoles. As for the DRM, is there an easy way to create back-ups of the games you purchase with these devices? Can you play them in any other way, than the way intended by the manufacturer? And would any of that be "legal"? I'd argue hardware based DRM is actually worse than soft DRM on PC games, as the latter is easier to circumvent.
dtgreene: * The controllers resemble the original ones, leading to a more authentic experience in that sense. I, at least, find it more fun to play retro games on retro controllers than using modern controllers (which are much thicker and have unnecessary (for this purpose) features like analog sticks) or a keyboard. (I am thinking of getting an adapter to allow the use of those controllers on my Linux PC.)
You can already buy all sorts of great, authentic retro controllers for PC devices.
dtgreene: * Couch multiplayer is easy on these devices. Unfortunately, the situation isn't ideal, as the NES classic only comes with one controller, and the SNES classic's multiplayer games tend to be competitive (Street Fighter 2), asymmetric (Kirby Super Star), or too long (Secret of Mana). (One thing to note: SNES classic controllers work on the NES classic, so there's that option.)
It is possible to hack these consoles, which can allow adding more games (though you lose the first advantage I listed), or even allow them to be used for non-gaming purposes (like music or video). Also, this can provide something for people to tinker with.
Couch multiplayer as well as customization/ added features/ mods etc. are also possible on a custom retro mini PC, which would offer a much wider range of options, without having to hack your way around artificial barriers created by the manufacturer. You also have access to a wider range of hardware and peripherals, and could use it as a true multimedia entertainment center, all in one device.
That leads me to another problem with all of these throw-away mini consoles: Lots of unnecessary electronic junk. Do we really need half a dozen of these things hooked up to our TVs, when a single device could offer the same functionality? And when you buy them for your kids, it is likely to lose its novelty appeal after a few months, and then it is going to be replaced by the next disposable toy.
I still think it comes down primarily to nostalgia ( these probably make fun presents, if, in example, your boyfriend or girlfriend is a "retro gamer" ), and convenience, because not everyone knows how to set up a custom retro station with emulators, suitable controllers etc.
That, and I think many parents buy these for their children, to show them the video games of their own childhoods.