I'm not sure "lust" is the right word. Didn't Tolkien say somewhere that Eowyn's feelings for Aragorn did not really change that much but that she grew to understand them better?
I'm not sure what Tolkien said on the matter, but I think you're right and saying lust isn't a right word. I think Eowyn was just misled by her own feelings in believing that Aragorn could be something for her that he couldn't. I think "unrequited love" would be a better term...ahh Shakespeare.