The following is my sincerest opinion (only with the academic career in mind).
Once upon a time, when our Profs were students, having a PhD alone could get you a job in academia; having one from a top university was probably a guarantee of a job at another top university. This is probably why we're seeing so many top unis staffed by people from other top unis at the moment. But those times are gone - for our generation coming up, it's going to be harder to get in but probably also a bit more meritocratic.
The name of the university per se does not matter. What matters most is the number and quality of your publications and the innovation you show in your work. What matters second-most is networking opportunities. Of course, it's not 100% meritocracy: while your individual performance matters above all else, people will also employ you if they like you and have heard from credible sources that you are good.
And this is where the name of the university sort of links in. The top universities attract the biggest names. That makes it more likely that you'll find a concentration of the kinds of people you would network with in order to boost your chances at a career. You will be working with them directly (your supervisor; your panel) and if you stand out they might mention you to their equally renowned colleagues over lunch or at the next conference. You will simply have more exposure to the top networks at a top university.
But, which is the top option can differ sharply from subject to subject. Oxford's and Cambridge's Politics & IR departments for PhD are not comparable in my view - having Cambridge on a CV might look nice, but for IR, you'd preferably want Oxford or LSE or Warwick or Aberystwyth. This is why I usually stress department, and not university. Hunt out your PhD opportunities according to where the best academics in your specific (sub-)field are located, and depending on the topic, this might not necessarily be a top 10 university. Ask yourself, which academics would I love to chat with about my topic X if only I had the right to knock on their office door? Then apply where they are.
If you are not at a top department for your field, you can try to make up for this by presenting your work at as many conferences as possible. You can also make up for this by publishing kick-arse stuff as early as possible (frankly, you should be trying to do this irrespective). Because a PhD student from the University of Average with publications trumps the PhD student from the University of Oxbridge without publications. Research university these days just don't give a damn if you can't help them get up in the league tables and/or get funding.
What I haven't yet been able to figure out is whether or not being located at a top department in some way relates to being more likely to publish - either because you're pushed harder to do this or maybe you have more freedom at a top uni (unlikely)? It wouldn't surprise me, but I don't even have anecdotal evidence of this.