Just how important is university prestige when completing PhD? I'm highly considering attending smaller, lesser known private university to pursue a PhD in cognitive psychology. The university gives full funding, I really mesh the with the professor I'd be working with, and the area is beautiful. The advisor I'd be working with came highly recommended to me by a professor I work with at my current university.

Compared to a lot of the other universities I've visited, the program is much smaller - with about 20 graduate students in total in the whole psychology department. This greatly appeals to me, as I just came from a tier 1 research university with over 60,000 students, and I've been really wanting a more close knit atmosphere. 

Ultimately, I think I want to pursue academia, although I'm open to the possibility of industry. My concern is that while the university itself is touted as a "hidden ivy", it's not well renown for its psychology department. The university is not a tier 1 research facility, and the department isn't highly ranked. I'm worried I'm shooting myself in the foot in terms of job prospects later on. How important is the prestige of the program in the grand scheme of things?
I guess that if you want to teach at Princeton (or Oxford) then go to a place just like Princeton (or Oxford). Otherwise, I think that what you do and the work you complete will have a much bigger impact on your future than the prestige of your institution, which may be overrated. There are a number of R1/Ivy graduates in the news that I consider to have the intelligence of rocks.

Having an advisor with lots of ideas, wherever they are, is a big advantage. Being able to work closely with your advisor is a big advantage in doing good work, provided the advisor does good work. 

And, your future may be determined as much by economic conditions as anything. You get no guarantees. In fact, even if you go to Princeton, expecting to teach there later, there is no guarantee that any suitable job will be available there (or at a similar place) at the time you need it. 

Choose your institution on a more sophisticated set of criteria than it's perceived reputation. 

I went to a big place with a lot of faculty and grad students. But the number of people doing anything that I was seriously interested in was very small. Three or four faculty and half a dozen students. 



Actually, I should note that many universities (not all) have a policy of not hiring their own graduates. This is to make it more possible to bring in new ideas from the outside, as well as spread their own reputation more widely. Some will make exceptions, but only for the most exceptional candidate. As an aside, I recommended that my own BS and MS students go elsewhere for their next degree, just to be able to bump up against fresh ways of thinking.