I absolutely love Lorelai Gilmore for many of the same reasons you'll see below, but my favourite female TV charcter has to be Veronica Mars. First of all, because she's just so smart - none of the helpless female heroine here, just bad-ass, down to business, tough Miss Mars attitude.
Also, Veronica's got this incredible ability to bounce back, no matter how many appalling curve balls life invariably throws her. It's such an inspiring thing, in a totally non-cliched way, to see someone take so many knocks, and for her to get up, a little damaged, but still strong enough to deal with it, every time.
The other most important thing about Veronica is her wonderful support network - she is so independent, and self-sufficient, but occasionally when things get too far on top of her and she has to finally look for help, she's not afraid to; even though Veronica's strong on her own, she's willing to ask for help and support when she needs it, and that's really something.