Is it possible that Jeane is lost. Whether it be in time or even memories? IF you ask me, in each game, she seems rather clueless and very...well, blonde.
I believe that she may have existed in the Ancient Armes Dynasty and that the event involving the Sun Rune caused her to experience something really traumatic that would involve her being zipped through time or losing her memories...etcetera.
And, notice, that in each game she is recruited in a Rune shop. Maybe all she remembers is that her fate involves a rune and by becoming a rune mistress she feels that she's able to get closer connected with her lost memories.
And, maybe...just maybe, that "Thing" or "essence" she got out of that casket was really just a piece of herself. A momento. An essence of her memory that was lost?
Also, it would explain why she took up the job for the Godwins. She wanted to get closer to the Sun rune?
But, then again, that wouldn't explain why she shows up in Suikoden I, II, and III if she found what she was looking for in suikoden V. Blah.
Thus being said, Jeane suffers from short-term memory loss