A fag hag is a term a lot of woman use... it basically means a gay male's best girl friend... the so called gay male in question is usually treated just like one of the girls and usually refers to himself as that as well.
The friendship is a straight woman with a very open and feminine gay male, and is usually a very close, best friend type friendship, in which you share lots of secrets, go shopping together, talk about fashion and beauty and relationship advice... so just because your cousin is gay, wouldn't mean that you would have a fag hag relationship with him. Just tell her you and your cousin aren't that close and she'll probably drop it.
You can see more info about it here:
http://en.wikipedia.org/wiki/Fag_hag
Now of course, there a lot of dynamics here and many people could be offended by this, but as hard as it is to believe, to the people involved in this sort of relationship, it is typically a term of endearment, though some do prefer more socially acceptable terms.
Also, not all gay males fit into the 'fag hag' category either, it's just a friendship style, usually one based on silliness. Not all gay men are silly, clubbers and effeminate, just like they aren't all cross-dressers either, so for her to assume your cousin would even want a 'fag hag' buddy is weird and shows she doesn't fully understand the term. However, to use the word 'fag' alone would be derogatory indeed.
How do I know so much about this? Well, I'm a hair stylist as well, and had a very best effeminate gay male friend, who people referred to me as his fag hag, we went dancing together, shared secrets, swapped advice, he dressed me up and did my hair... actually, he called me that as well come to think of it! Sometimes, I kind of miss him.
ps. some women use terms like this to appear like they are in the know and take it as a joke... it sounds like these woman are doing that since their point of reference is Sex in the City. Kind of poser-ish if you ask me!