Wait, difficult child is going to contact the dead kid's mom, or your neighbor is?
To play Devil's Advocate, I don't believe in ghosts/spirits hanging around post-death. I do believe in the power of suggestion, mob mentality, weird old houses (used to live in one, people thought it was haunted - we proved scientific reasons for each and every weird thing that happened in it - probably why I don't "believe" anymore), and totally freaking yourself out.
I guess my real point is how AWFUL it could be to contact his Mom, no matter who's doing it. He was a difficult child? If she DOESN'T believe in ghosts, you've just reopened an old, painful wound. If she DOES, now you're making her deal with her difficult child, even after he's gone. How many of you would want to have to put up with your difficult children for eternity??
Sounds like a lose-lose for her, either way. I would NEVER want someone (even a family member) to call up my Mom after I'd died to tell her my spirit is restless and stuck in my old house!!
I respect all the beliefs of the people posting above me. So don't read what I'm about to say as ANYthing other than "have you thought about this....". NOT judging, just giving my advice!
I'm just saying to consider (because it's always good to consider all options!) the possibility that one by one, your family has gotten so worked up about this that you're now feeding off each other's fears and making the situation worse. Your kids already have anxiety issues - I can't imagine adding "omigod our house is haunted" fears into that mix. You're all under a ton of stress with the move, work, the mice, the uncaring landlord, and parenting a difficult child to boot. I could see how that, plus a new house that is old and does weird stuff, plus finding out people died there could equal "ghost!!", whether true or not. And the kids could have started messing with you as a joke (Ah, difficult children), and freaked themselves out, and now believe it, too. Seen that happen before.
Not sayin' this is what's going on. Just some stuff to consider, as an alternative.