I'll try to make a first little attempt at parts of an out-of-universe answer why the writers might have chosen Saturn, even if it might be a bit far-fetched or speculative.
Aesthetically as well as regarding many small references the movie is largely indebted to Stanley Kubrick's seminal work 2001: A Space Odyssey (well, which movie of this kind was not influenced by that, but there are some rather obvious references), where signs of extraterrestrial life emerge from near Jupiter. But in fact in the original novel by Arthur C. Clarke it was indeed Saturn (or one of its moons) and this was only changed to Jupiter in the movie because credible images of Saturn were simply not producible at that time. So this might very well be a reference to 2001, albeit a slightly distorted one. And in fact there seem to be many precedents of Jupiter and Saturn serving as gateways to new experiences in science-fiction works, though it's unclear if that all goes back just to 2001 or to some earlier precedent, or maybe a more general affinity of human culture to those planets (or if all the other planets are equally represented). But I guess being such major players in our solar system and way farther away than Mars (which is largely demystified nowadays, the times of Mars invasions are over) makes them a bit more predestined to serve as such gateways.
Then there are other small hints that Saturn was a fitting choice, even if they're way more far-fetched. First of all Saturn is the Roman god of agriculture, which could on the one hand represent the largely unprogressing society of farmers humanity has accomodated itself to. When then factoring in the meaning of the Saturn Rockets as the facilitators of the Apollo missions and thus a signpost of human space exploration, it could on the other hand represent the pioneering attitude with which Cooper, the compulsory farmer and former space explorer, grasps for new directions in space (and through him ultimately humanity). But I guess culture is littered with other Saturn references and one might find fitting analogies of varying degrees.
In addition to this, Saturn might as well have served as a visual counterpart to Gargantua. Showing Saturn with its commonly known and characteristic rings could have served as a visual clue for the audience to understand that it is the black hole's accretion disc, a rougly similar ring of stuff (albeit faster and hotter), that gives this black hole its at first rather unexpected appearance. Gargantua is basically the "weirder" mirror image of Saturn.
Those are really just some first thoughts and this whole matter deserves way more investigation. I also admit that I don't currently have an actual in-universe answer for "them" to place the wormhole at Saturn instead of a nearer location. But it seems irrelevant anyway, given that any other planet might not carry the same aesthetic significance for the story as Saturn does and that placing it directly in earth's atmosphere and the atmosphere of Edmunds' planet (as suggested in a comment) would result in an entirely different story lacking any kind of interstellar travel, time dilation and time-defying communication (apart from other possible dangers a wormhole right in the atmosphere might pose). But maybe the safety argument still applies to a larger scale. Afterall this thing has sat there for 48 years already and it is hard to anticipate what stuff may come through it or in which way it might have some extraordinary gravitational influence, so better keep it at a safe distance.