It could be a way of taking over someone's email account, even if they are alive. It would not be difficult to fake an obituary, especially since anyone can create an account and post a short blog. Most of these sites require a photo of a death certificate, but those are also easily photo-shopped.
Imagine what sort of damage you could do if you had access, even for a day, to Tim Cook's email? Some other CEO of a Fortune 100 company? A celebrity?
I'm surprised this isn't used more often as an attack vector.
You might not care, but there could be financial ramifications for family members.
I had to dispute a charge to my wife's credit card after she passed... it felt like the credit card company was fighting tooth and nail to make me pay for a charge that I hadn't made.
Please provide an encrypted file with all of your emails, pictures and videos of you (including any you might be naked in), scans of any letters, birthday cards, etc you might have. And then post a link to a service which will provide the encryption key once you die.
If you have adult children/family, make sure they do the same.
and obviously other people can be affected by the impersonation of someone, e.g. an impersonator taking advantage of a deceased person's trustworthiness within a community that is not yet aware of their passing.
Why would you care? You'd be dead...