There are few things more challenging for a family than losing a loved one. Historically, that person's digital existence was of lower importance relative to everything else going on during the painful transition. However, as more vital processes move online, the problem of a next of kin in the digital world will need a cleaner solution.
Right now, your best bet is to use a password manager and make sure your partner or chosen family member/friend has the master password. This isn't the worst choice, but most people don't even do this much. Furthermore, it does nothing to address the wishes of the deceased. Do they want accounts deleted? Ignored? Made into memorials?
A more thorough thought experiment would be required to design a complete solution, but this is touchy territory. Aside from the dark nature of the topic, there is risk in any single company's solution here. If someone provides a "take care of your digital existence" solution, what happens when they don't get that last payment? What happens if they go out of business? That risk holds true any time, but when the originator of the request for services is no longer available, it passes the burden to someone who might not be equipped to handle it or fully understand the nature of the situation.
An open standard might be the solution here. On top of the standard private companies and services can and should be built, but this would allow you to not get totally locked into a single service for example. Companies could try their various sales models, "free", cheap but minimal, expensive and full featured, subscription, etc.
It's an interesting discussion to have and I suspect will eventually be a thriving industry. Only time will tell how this problem is solved and what services flourish to meet the needs.