Letters to the Editor
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As someone who's done the chemo
It's not melodramatic of her to mention it casually but not linger long on the subject. You already said she treats it lightly and she probably isn't sure how much you do or don't know. She doesn't want to keep it a "Secret", yet she doesn't want to treat you like an idiot nor shove it in your face.
SIMPLY: It's a really awkward topic - to tell people you've got cancer (or whatever she's on chemo for). In fact, a friend and I called it the "C-bomb", referring to its effect on any social conversation.
I wouldn't say you have to avoid any mention of the idea that you're naive/scared and don't understand what exactly her illness means in terms of whether or not she's terminal...but don't linger on the idea of death any more strongly than that. Otherwise Cary has it right.
Your friend might be happy to "teach" you about her illness, once she knows that you don't feel informed. Spreading awareness is a way that many patients feel that they are doing something "useful" through their illness (for real lack of a better way to say it).