Whovian

Hi Whovian!
I'm not in anthropology either, but my experience and knowledge (in biological sciences) tells me that finding a great mentor outweighs the potential prestige of a program. First, grad school is tough and having a great mentor will go a long way. Second, while the name of a program may carry a certain amount of weight your grad school/research output, mentor recommendation, physical/mental well-being carries more. Thirdly, you always have the option of establishing connections/relationships with POIs in prestigious programs without necessarily attending the program.
I hope this helps! Although it seems as if anthro is quite a bit different, anyways that's my two cents!

I don't know whether to take an offer from a "not top tier, not well known, but not unknown" department. 
The funding is hand to mouth, but everyone there tells me they are happy. I couldn't possibly "fit" better in too many other places. I idolize this advisor too--great work, great experience, happy students. But I cannot shake off the nagging feeling that I will never be able to compete with TT jobs with PhDs from famous places. 
I have been a star student all my life. But do not know how to choose a grad school -- what to prioritize and what to ignore. I don't want to be jobless five years later
 

Forget rank.  Rank is all about ego, really.  Unless you're hoping to be in academia.  For professorship, it is all about what school you went to.  But, your research is most important.  Companies don't care about your school, but rather how good your experience is.  If you school was the only thing that mattered, hardly anyone would have a job unless they went to Harvard or CalTech.  If it is a pride thing, just swallow your pride and get into the school that is clearly the right fit for you! ?

I don't know whether to take an offer from a "not top tier, not well known, but not unknown" department. 
The funding is hand to mouth, but everyone there tells me they are happy. I couldn't possibly "fit" better in too many other places. I idolize this advisor too--great work, great experience, happy students. But I cannot shake off the nagging feeling that I will never be able to compete with TT jobs with PhDs from famous places. 
I have been a star student all my life. But do not know how to choose a grad school -- what to prioritize and what to ignore. I don't want to be jobless five years later
 

It's hard to maintain perspective in the face of rejections, but just look on the bright side -- if we don't get in to our chosen programs we are probably on track to be ahead financially and practically in life over those who sacrifice six or seven years at poverty level fellowships and stipends. I applied to seven schools, interviewed at one and then rejected from that school plus two others. No word yet from the other four, but implied rejections from two of them are expected. I'm not hopeful at this stage and ready to move on with my life. Graduate school is not the only path to happiness and knowledge, just one of them.