I don't think there is any genuine liability. What is the family going to sue for? "pain and suffering"? Couldn't they by the same standard sue for that now? No, Yahoo could only get sued by the executor of the estate for breech of contract. What financial loss is there? None, it's a free email account.
Not to mention, just have the tech purge any nastiness anyway, and be done with it. It's really that simple.
Now they'll be haunted, harassed and pilloried by this family lusting for things they have no right to all out of grief. Instead of being thankful for the time they had with their son, or for those of his peers who chose to speak to them they aren't satisfied. There's no unquenchable appetite like grieving parents.
There really is nobody left who could possibly give a hoot about the contents of my email box. But I think I'll still add a codicil to my will that said contents (as well as my hard drive) should just be allowed to vanish into thin air.