It all really depends on what you mean by “prestige” and how you think you might use it.
In academic and governmental circles, all over the world, the University of Chicago and its faculty are seen as in the very top rank of universities world-wide, absolutely a peer of Harvard, Stanford, Berkeley, Michigan, Oxford, Cambridge, ENS Paris, whatever university you want to name. Obviously, that does not extend to traditional engineering – which does not exist at Chicago – and no university is strong in every department. Having a faculty appointment with tenure at Chicago, or getting a PhD, JD, MBA from Chicago – that’s as good as it gets in Elite World, at least for people acknowledging the quality of your credential and your accomplishment.
In terms of what an undergraduate degree from Chicago means in Elite World, things are more ambiguous. On the one hand, everyone knows that Chicago ABs and SBs studied with that great faculty, and moreover that Chicago has a unique curriculum and culture that make it probably the most uniformly rigorous non-technical university college in the country. On the other hand, everyone also knows that for decades Chicago did not compete very effectively for the flashiest students coming out of high school, that its admissions criteria took no account of things like leadership qualities, and that it did not turn away many qualified candidates. So for people older than, say, 28, simply having been admitted to Chicago means a lot less than having been admitted to HYPS or MIT in terms of how impressive your “natural talent” is likely to be. Having an undergraduate degree from Chicago is a little like being a Marine: we know you’re tough, we know you’re well-trained, we know you have great skills and great values . . . but if you’re so great, why the heck did you enlist in the first place?
In short, a Chicago bachelor’s degree doesn’t support an inference that there’s a 99% chance you are a total winner to the extent that a degree from Harvard or Stanford does. People feel the need to check you out a little further. But that doesn’t mean that Chicago alums don’t compete successfully with Ivy alums and the like in the real world. They do; they get those opportunities, the second and third interviews. Moreover, Chicago has changed its admissions practices and has become very nearly as selective as the most selective of its academic peers. That is pretty widely understood in Elite World, and will become more widely understood if it is sustained over time. Ten years from now, an AB from Chicago will probably have about the same signaling function as an AB from Harvard.
Outside of Elite World, prestige comes from four sources: football, basketball, Hollywood, and, to a much smaller extent, USNWR. Chicago essentially doesn’t play in the first three arenas at a level that anyone would notice. That’s not likely to change anytime soon. It’s also in the Midwest, at considerable distance from the places where most prestige seems to be mined: the Amtrak Corridor and coastal California between Marin and Orange Counties. That’s also not going to change. (Note, however: I know two first-year students at Chicago from LA. One is the double-Ivy-legacy child of a Hollywood executive, the other a grandchild of a longtime board of trustees member at a single-initial university. Both families are thrilled that their kids are going to Chicago. But both families are solidly rooted in Elite World.)
Anyway, what is “prestige” good for? Getting yourself a hard look from a top PhD program in your field, or the Rhodes Scholarship selection board? Chicago is just about the best there is for that. Professional school admissions? You’ll get respect, but they tend to be much more numbers- than prestige-driven. Fancy job opportunities in Elite World? You will qualify; your foot will be in the door. But then like all the other elite college grads you’ll have to earn it. Post-docs, fellowships, clerkships? It’s maybe a little burnishing around the edges. My wife and I are both Yale summas, and we would be hard pressed to say that has made any difference in our careers after we turned 25.