IMO Android tablet is still the best bet, especially if NDS is on the table. That makes it the only option other than an actual DS/3DS.
But under $100 is a tough target. You can get a Hisense Sero 8 for ~$80 on Amazon, which is what I usually recommend for "cheap" tablet gaming, and that leaves around $20 for a controller. Ipega PG-9023 on amazon is usually ~$23-$26, which hits your target almost on the nose and gives you a setup where you can wrap the controller on the tablet like a handheld. That's not everyone's cup of tea but I like it and do that with my Shield tablet.
This Astrotab also looks good for the price but I don't know anything about the brand, so caveat emptor.
Both the tablets I linked have at least 1.5GHz and 1.6GHz SoCs, so good enough for NDS, and maybe even some PSP games. The Sero at one point used the Tegra 4 (I think), though I don't know if it still does.