I played Sacred Stones only once so I can't exactly remember all that much of her character (as such it seems the moment she gave Lyon the stone didn't strike me as entirely idiotic) but her naïveté really did give me this "cute but rather tiring" feeling, so I get where you're coming from.
However, I distinctly remember liking Eirika. For one she's a female character - and that might just be me, but that's already a good thing in my book. Like many said before, her personality and her realistic growth also gave me something to get attached to. I tend to like characters who act upon their feelings, even if it means I get to sit down and watch something frustratingly annoying happen once in a while, because I do stupid things just by acting on what I feel myself. As long as I could relate to her (i.e. the whole game), I felt connected to her and it made me want to continue to see how her story ends.
She was naïve and was one day suddenly thrown into war. For what it's worth, she did a pretty good job at managing a party with this lack of experience and I respect that a lot.