I read through the whole thread, and there are a lot of good suggestions. I thought Fox was clever. Firebird is a logical choice, but to me it seems a bit over-used. I loved the symbolism behind Athena, but I would not be comfortable naming a browser after her. A database system, maybe, but not a client program. I'd like to throw in another suggestion though:
<strong>FLARE</strong>
<a href="http://dictionary.reference.com/search?q=flare">http://dictionary.reference.com/search?q=flare</a>
It's short and pronounceable, keeps with the fire theme, suggests its homophone 'flair', and has connotations of speed and light. It seems to fit well enough with Thunderbird, too, particularly since lightning can be described as a flare.
As for being taken, the TESS reports a trademark for computer hardware for descrambling and decoding, and software for the engineering design of process safety flares. There are two somewhat relevant trademarks for telephone equipment and PDAs, and software used in controllers and data acquisition systems. However, both of those are abandoned.
The first 15 pages of a google search for 'flare -pants -"solar flare"' yield an <a href="http://www.flare.net">ISP</a>, a <a href="http://www.xara.com/support/docs/webformat/spec/">graphics format</a>, and a <a href="http://www.flare.org">computer language</a>. Flare.com, flare.org, and flare.net are all taken.
'Flair' itself is not trademarked in the US by anything remotely resembling a web browser. (I think I only saw one software trademark on the entire list.) flair.org is for sale, and general consensus seems to be that Phoenix definately has flair:
flair n.
1. A natural talent or aptitude; a knack: a flair for interior decorating.
2. Instinctive discernment; keenness: a flair for the exotica.
3. Distinctive elegance or style: served us with flair.
The first 10 pages of Google reveal a sort of <a href="http://healthpolicy.stanford.edu/PALS/program.html">interface</a> for the elderly, a German (?) <a href="http://www.flair-fem.de/readme.htm">design program</a> and another <a href="http://www.rola.com/flair_e.htm">software</a> for police case processing.
Anyhow, that's my 2 cents. Y'all have fun picking a name.