The reason the invisible lady is invisible is because of a programming bug in the NES version of FF1. At least, I assume it was a bug; it could have been intentional, who knows. Normally, people's sprites are visible most of the time, and when you enter a room, they become invisible. You can see this when entering any room; the people outside disappear, the inside of the room becomes visible, and usually the outside darkens a bit, to simulate being in the room. Well, with the Invisible Woman, her sprite was accidentally made visible only when you're INSIDE a room. If there was a room within site of her, when you entered the room, she'd become visible. But there is no room close enough to her to let you see her. To solve this problem, we must use the Walk Through Walls Game Genie code.

To use this code, go into a room in Coneria Castle. Stand near the bottom of the room. Open the ROM Patch window in Nesticle from the menu, enter the code SLZIAA, and double click it to activate it (it should be flashing). Now, whenever you open and close the Menu screen in the game, your character will walk down one space, walking through walls, over mountains, even onto the ocean if that's the next space below him. Use this twice while in the room, and you'll be outside the room, but with the insides of rooms still visible.

Walk up to the Invisible Woman, and lo and behold, there she is, in all her glory. You'll notice that now, everyone ELSE is invisible. At last, the mystery is solved.

Of course, in case you don't feel like trying this yourself, I've taken some screenshots.

...through the wall. Now to visit the Not-So-Invisible lady one more time.

Unfortunately, she still has nothing new to say.