You got a new Nikon DSLR kit for Christmas, congratulations. You've got a little money left over and you want to make the one upgrade that has the most bang for the buck. For the consumer cameras, the answer is easy:
- DX DSLR with kit lens: US$200 will net you a discernible image quality difference. Just buy the Nikkor 35mm f/1.8G DX lens. Given that all the new DX cameras are 24mp, the kit lenses these days are struggling against two factors: first, they're complex optics designed to be inexpensive, and thus not exactly the resolution champions that a 24mp sensor needs to show off what it can do. More importantly, stop down a stop or two on the kit lenses to get to their sweet spot and you're going to throw some diffraction into the mix. The 35mm f/1.8 DX lens is very good optically, and at f/4 it's about as good as it gets short of anything other than an expensive, exotic lens. Bonus: that f/1.8 is a stop-and-a-half faster than the kit lens, so in low light you don't have to boost ISO and pile up noise.
- D610 with kit lens: The same thing applies here, though you get some options: get one of the f/1.8 prime lenses. At a bit above US$100 the 50mm f/1.8D is still available, and from US$275 to US$700 you have 28mm, 50mm, and 85mm options to consider. As with DX, you'd also be getting a lens that is faster than your kit zoom, so can keep the ISO values a bit more in check. All of the f/1.8G options are very good, and the f/1.8D options are still very respectable, as well.
With the higher-end FX bodies (D800, D4), they don't typically ship with a kit lens, so the quick and easy upgrade option might not be available to you. That said, if you're trying to use a variable aperture zoom on the D800, having any of the primes available is an optical upgrade at that focal length.
The Df and D4 are more curious beasts when it comes to lenses. The 16mp sensor isn't going to stress optics all that much. The f/4 zooms (16-35mm, 24-120mm, 70-200mm) all perform exceptionally well on this sensor in my experience, so if you're missing any of the wide angle, midrange, or telephoto zooms, that might be a good upgrade for you. Even the two recent variable aperture zoom updates (18-35mm, 24-85mm) perform quite nicely on this sensor, though they probably wouldn't be an optical upgrade from what you're using, just a convenience upgrade to make for a smaller, lighter package at times.
If you haven't gotten the hint, I'll state it more obviously: the one place where you're likely to get a tangible benefit above and beyond the new camera is in the optics. My second choice would be to upgrade your support system, especially if you've moved to a higher end body (e.g. gone from light DX consumer body to heavier FX body).