TadMack at Finding Wonderland asks, "Wouldn't you hate for someone to be digging through your computer files for all of the story fragments you've begun and abandoned, to be published posthumously?" The question refers to the announcement that J.R.R. Tolkien's son has completed one of his unfinished manuscripts.
Speaking as an author who has a son who could easily end up a writer, I would love to have him finish one of my works. It would be as if the two of us got to work on something together, if that's not too creepy for you. I wouldn't even object to his taking an unfinished piece or an idea and giving it a "next generation" treatment. He could go ahead and let the piece evolve so it fits the culture he was then living in, even though I wasn't. It would give my work a new life and bring it to new generations.
Go for it, sweetheart.
Of course, this is all assuming he would do a good job. Otherwise...