This question has been asked before on other forums, but I feel it might be profitable to ask it here. What if Padme survived? How does the Dark Times and the ensuing Rebellion change due to her presence?
How?: Lets say Obi-Wan, Yoda, or both used the vital transfer Force power and saved her life, perhaps at the cost of one or both of their own.
She paints a big, giant target on her kids, that's what happens. Even if she went into hiding Vader would never leave her be and he'd know if she was alive. And when he tracked her down he'd know she was lying if she tried to tell him the children were dead (assuming she didn't keep them). Honestly, dying was her smartest move and likely why she didn't try harder to live. Her death provides the perfect cover for her children and keeps them safe from the Emperor.