I'm not a lawyer, but here are the questions I'd ask, which likely have different answers for different online services, different states, and different countries:

1. Are the rights under the EULA / ToS transferable due to terms explicitly in the EULA governing transferability?

2. If there are no explicit terms, what is the "default setting" for transferability?

3. Are the digital items being sold considered goods, services, or some combination? (I'd imagine it's much easier to terminate a contract for ongoing services than to negate heirs' rights to goods the deceased has purchased.)

4. Is there a catch-all "escape clause" in the EULA which says "We're allowed to terminate your right to access these songs or movies at any time for any reason, or no reason"?

5. What about the first sale doctrine? Doesn't it say that you have some right to transfer items regardless of what the EULA / ToS says?

From a practical standpoint, you can probably transfer the account by just giving the password to the receiving heir. Making new purchases on the old account may no longer be possible if they check to make sure the name on your credit card matches the name the account is registered to, but I'd imagine many providers don't do this check and only care about what your CC info says if they start getting chargebacks.

If you don't know the password and don't have access to the deceased's email to use the usual password reset mechanism, you may be facing an uphill battle.

It would also be a micromanagement nightmare to have several accounts on the same service with different content. It might also trigger provider's alarms if you access several different accounts from the same IP address (although if the number is reasonably small -- less than ten, let us say -- hopefully they'll assume it's legit, i.e. multiple accounts belonging to different people living in the same house with a single NAT'ed Internet connection.)

The simple fact of the matter is that most Web businesses' domain modeling assumes that customers live forever. What happens when they don't is basically anarchy. We're still at the point where most of the people who die never had a large online presence, but in 30-50 years that will no longer be true.

An ideal situation for content providers would be for all content to be destroyed when someone dies, because then the next generation would have to buy it if they wanted it instead of inheriting it. The problem is that this leads to cognitive dissonance, because heritability of possessions is a central feature of many cultures, including ours.

An ideal situation for consumers is complete transferability. The first problem is that this encourages piracy -- if there was a free and legal resale market where any song could be sold for (let us say) $0.50, while songs can be bought on iTunes for $1.00, iTunes would make exactly $1.00 on each song it released, because exactly one person would buy it, then resell it (keeping a local copy), and its buyer would resell it, etc. The second problem is that it's difficult for a company halfway around the country to differentiate between the legitimate heir of someone who's actually died and an account thief.