I'm not sure whether you're referring to your Adsense account or your Adsense earnings.
If it's the former, you can certainly bequeath websites and other online assets to another person - but as far as I'm aware, Adsense accounts are not transferable. Your family member would have to apply for their own Adsense account and apply it to your sites.
If it's the latter, your executor or probate lawyer would need to apply to Google for access to unpaid Adsense earnings. If the earnings are paid into your account then they become part of your estate.
I have done something like this. About 10 years ago I was making over $100K USD with Adsense. The tax burden was too much on me so I had my woman apply for an account. Back then the processing was pretty fast, about 10 days. At the time I only had one large site so on half the pages I put her code and the other half mine. Now I have a handful of smaller sites, she has a few and I have a few plus our jobs. In the future you could migrate your heir's code into all the ad slots and the money would flow into their bank account. I don't know what the wait time is these days or the details of getting a new account. I am sure it is harder and the wait longer. You would need a new established standalone site to apply with but once accepted you can move the code where you want.
I've helped the families of two publishers with this in the past. In one case the original account owner had made arrangements before their death, in the other they hadn't.
The publisher who had made arrangements moved everything into a limited company with next of kin as a director. Obviously there are some drawbacks to that, but it worked well for them.
Where no arrangements were made in advance we (as a Google Certified Publishing Partner) worked with Google and the family to help get the issue resolved. It wasn't too difficult, but it was fairly time consuming.
If your account is a personal account in your name I would definitely suggest having some sort of plan in place. Even if that is just that funds get paid into an account that kin can access and they have logins, that is a big step.
That will depend on the structure of your business and the corporate and probate laws of your country.
My business operates as a proprietary company with my wife as the only other director. If I die then she takes over ownership and control of the company. She can then operate or dispose of it as she sees fit.
You can do this without changing anything at Adsense.
If you currently receive Adsense payment in the form of a check, open a small account at any bank and add your family member to the account. They could deposit the checks, wait a few days to clear, then withdraw money.
If you currently receive Adsense payment as direct deposit, just add the family member to the accout so they can make withdrawls. The Adsense account can remain solely in your name even after your death. Just don't tell anyone :)
Note: there may be tax issues at some point in the future depending on your location.