A top university. Just like scientific journals, these organizations sit at the top of a prestige hierarchy that makes them difficult to dislodge, and everyone wants to get into the top schools/journals to show how smart and capable they are. A university or journal with a strong EA agenda could do a lot to change elite culture, by unilaterally changing the target that everyone else is aiming for. (Of course, even if EA owned a university like Harvard 0r a journal like Nature, we wouldn't want to completely change the target because that would probably mess things up, as pradyusp mentions in their comment. Rather we'd just want to nudge things in a marginally more EA direction. But this could still have huge effects in the long run.)
University of Chicago, and places like the Hoover Institute at Stanford, are perhaps examples of top universities that take a different cultural stand (slightly more conservative / libertarian) compared to the leading Ivy League schools. I think the existence of U Chicago probably has an outsized role in supporting the prestige of libertarian ideas in elite circles. (Aka not very prestigious, but imagine how bad it would be without the schools!) Alternatively, EA could buy a medium-prestige school and convert it more wholeheartedly to embracing wild EA ideas and priorities -- George Mason University is an example of a school with an extremely iconoclastic libertarian department that has an outsized influence on public thought relative to the school's overall prestige.
I think buying a university (more realistically: funding a new EA-themed department within an existing college, like the Hoover Institution or GMU Econ department), or a top scientific journal in a key field, would be more impactful than buying a newspaper like the Washington Post, which is a potential EA "megaproject" I've seen tossed around before on the Forum. Newspapers are more about mass political power rather than influencing elite & academic opinion. And the news business is much more competitive -- the likes of WaPo and NYT, although powerful, are less unassailable fortresses of cultural command in today's world than they were in the past. By contrast, the dominance of top schools and (perhaps to a lesser extent thanks to open-access & sci-hub) top journals, seems to be going strong.
EA is already obsessed (rightfully so I believe) with starting effective-altruism student groups at as many top schools as possible. Eventually, funding some schools to set up formal academic departments inspired by longtermist moral philosophy, scientific analysis of potential existential risks, EA-aligned economic development economics, or etc, seems like the logical next step. Just like Blackwell books going through financial trouble, maybe EA could keep an eye out for an opportunity to bail out a financially-distressed but otherwise promising university, and thereby pick up a lot of elite social influence at a bargain price.
I like the idea but then a potential counter-point would be we should just simple expand Oxford's GPI/FHI. Both of these are within Oxford, which adds a lot of prestige/credibility etc.