I suppose I could be cutesy and say, "Why not a ginger cat?" But it looks like I did, in fact, offer to answer the question.
Well, I suppose it all started with the Yiddish word gingi, which means redhead. When I was a very small child (three or so), my parents had a friend with a very red beard, who was called Gingi. He was the first person I'd ever met with hair that was similar in color to mine. And yes, Jewish redheads are fairly common; my friend Laurie, who's also a redhead, posted a FAQ about it.
On my third birthday, I was given (perhaps by the same person) a plush giraffe of approximately the same color as our hair. I named him Gingi as well. Eventually, Gingi the Giraffe was relegated to a basket of outgrown toys, but I always remembered his name. Now my niece and son play with him. My son even has his own Gingi Giraffe, a lot sillier-looking than mine.
Later, in college (at Brandeis), I encountered the problem of Too Many Rachels. There was even another Rachel Silverman. But nicknames weren't common, and so I was just the red-headed Rachel. No, no, the one with the really bright red hair. Yeah, that one. (You see the problem.)
Then I got on the net and started thinking of possible nicknames. I didn't have to think of a username while I was a Brandeis student, since they assigned me one (st883612, if you're curious). I was the only Rachel in alt.callahans for a while, and my character also carried around a green firelizard named Soora. Then my (now ex-)husband gave me a mail account on his UUCP machine, with the username "carrot." When we broke up, I started looking around for a new net identity, as "carrot" held too many memories of private jokes. I really wanted gingi, but it was unavailable on the first two ISPs I tried. So first I was "fuega" (on Netcom), then "cinnabar" (on another ISP that has since changed its name several times), and then, finally, "gingi" when I got my panix account.
And then I got back into SF fandom, and people started calling me gingi in real life (always having it on my con badges probably doesn't help). So it became a name I identified with, though I still automatically introduce myself as Rachel.
But there are other people who identify with the name gingi, and periodically I would register for something on the web and find that it was unavailable, or that I was required to provide a longer name. So I started using gingicat in those instances.
At one point, my friend Zev (an Australian Orthodox Jewish filker) overheard me explaining for the nth time that gingi means redhead in Yiddish, and he said, "You do know that was actually stolen from British English, not German or Hebrew [like most Yiddish words], don't you?" I thought about it, realized that most British books refer to redheads as being ginger-haired, clonked myself on the forehead, and said, "Duuuh!"
And then, sometime in 1999, my friend Merv and I had long email conversations, a large bit of which was the imaginary conversations and adventures of Phhbbbtttt the System Dragon, the Ginger Cat, and Dayglo the baby bear (photo taken by Merv at Baitcon). And so the persona of the Ginger Cat finally emerged.
Later on, I got persuaded to go on IRC, and decided to use the nick gingicat there as well. It's fun to pretend to be a cat around my friends.
And then my friend Neil was throwing out old puzzles and found the picture you see above... and several people who were present exclaimed "That's the Ginger Cat!" Originally, this website was called Rachel's Nest, but that picture was so perfect that I redesigned the entire site around it. And so it goes.