6 months seems to be cutting it pretty close - as a general rule, everything takes 2-3 times longer than you expect it will. Remember, there's a good chance your first launch will fail. We launched 2 sites in the 9 months where I kept my day job, and both went kerplop. That could be because of a half-hearted effort (the app I'm working on was also the main plan, I just couldn't concentrate enough to build it while I still had the day job), but in general easy sites don't make money, and sites that make money aren't easy.
Also as a reminder, the more you have achieved, and the more you have to show, not only the easier you can get funding, but the higher you will be evaluated, which is definitely to your benefit. I would say giving out shares and raising money just so you can pay for your life expenses is not the best idea, get a line of credit!
Anyways, it looks like you are in better shape than I thought. All the best to you.
But I would say in general you would spend about 6 months developing your product, and having something ready anywhere from a good working alpha version to something more of a working prototype and work in progress. I hardly doubt there are too many companies that can start generating revenues (let alone profits to spend on living expenses) within six months from the day they start.
So I would say the best way to do this is to start working on the idea before you quit your job, try to get a prototype or something together, and start looking for funding. I think in your case at the end of six months you will def need outside money (again since I doubt you'll be generating revenues by then and you are out of savings).
So regardless, if you do quit, I would recommend you looking for money right away. The worst thing to ever do is to ask for money when you need it, and when you are under pressure. Take your time, do your shopping, find out how interested people are in your product and try to raise some smart money before your time is up.
and please do keep us in the loop, and good luck to you