So I had a thought, and figured I'd run it by everyone here before putting it into practice:

In Light's introduction post, it was implied she'd killed a rookie from the academy. I was thinking it'd be interesting if Kaede knew that 'newbie' and has a reason to hold a personal grudge. Whether or not the Gotei 13 (and Kaede herself) knows who the killer was doesn't matter so much; it could still be an unresolved incident, but it'd give something to build off of.