Saving Throws have already come up several times in my new No Attack Roll Combat System. Please everyone check out this link. It is SHORT (compared to my other 3-5 page diatribes), and explains my understanding of what saving throws actually represent.
And IF you buy my ideas, you can see that it's not necessary to have "double saving throws."
What do I mean by a double save? Well, if a monster needs to get a successful attack to cause a "special effect" (like a cockatrice bite, poison spider/snake, whatever), AND the character can still avoid it with a saving throw, you essentially give the character TWO saves...the first not determined by class or level, but rather by AC.
Which is probably why fighters turn out to be biggest tanks in the game...but I digress.
I don't think we need to give everyone a double save, personally. Now, there might need to be some adjustments to the saving throw matrices if you think things should be EASIER for the PCs. But in general, I think they're pretty good. Maybe we need to make poison a little more "slow-acting" (as in real life) to give characters a chance to suck wounds, or induce vomiting, or cast neutralize poison (i.e. "the less gross method of treating poison").
All right, hope that helps!
[By the way, I realize I have a ton of post on this blog (did I mention I just broke 500? Oh, yeah...already bragged about that), but there's a lot of juicy ideas buried in there along with my rants against killing people in the Middle East. Skim 'em when you can...some of 'em are fun!]