Would the Christians Luke was writing for know who Phanuel was? I'm sure there are many more names in the New Testament that no one would know, but this one strikes me as very odd.

The prophetess Anna is described in pretty good detail, for the Bible at least, but she doesn't do anything.  She is described as:


  the daughter of Phanuel, of the tribe of Asher. 


Does his name give any credence to her story or her prophecy?
I don't think you'll get a definitive answer on this one.  There's no other reference in Scripture, so there's nothing to cross-reference.

My hunch is that some people would and some wouldn't, the same way some people in my Church would know who Ray Comfort is, and some wouldn't.  

It might simply be the habit of recording familial lines to avoid confusion with another Anna - this was Anna, the daughter of Phanuel, as opposed to another Anna.  

I think that we have to assume that at least some (if not all) of the intended audience either was familiar with this person, or would be able to use the familial reference to determine the identity of this person.  

Identification of a person as the "son of" or "daughter of" another person is a relatively common method of distinguishing individuals, not just in Biblical times, or even just in the Middle East.  I've heard references like this from Scotland, England, and various other places, both in historical works and in literature.  (I've even heard it in science fiction  i.e. Worf son of Mogh.)  

I believe that this was merely a common expression that most of the intended audience wouldn't think twice about.