||For a PhD, your advisor is the most important thing. Name recognition of the school does matter, but if you pick a bad advisor at a good school, there's a good chance that you won't finish, in which case the name of the school never does you any good.
Having said that, if you're like most people, you probably won't really know whether you like your topic or your advisor until you get started. Therefore, all other things being equal, it's better to pick a place that has multiple people and topics that you think you could work with than a place with only one potential advisor. In a similar vein, look for a school in which it is possible to change advisors. In most places it won't be easy to change advisors, because your funding will be tied to your advisor, but some places make it easier than others.
One last note. I would avoid anyplace that has a model of "weeding out" their students. Some places/projects bring in more students than they can support in the long term, planning to drop some of them along the way. It's one thing if you decide on your own to leave, but you don't want to get a year or two in your program and be told that you have to leave because your project can only support the top N of their students and you didn't make the cut. Of course, such places aren't likely to tell you that that's the way they work ahead of time, but you can probably get some idea by talking to the current grad students.