Question: What kind of fallacy would this be?
So here is my take on this:
Mother: No, I won’t let you drink energy drinks. There have been cases of people dying of heart problems after drinking them!
First, I give the mother credit for giving a reason! My mother was a big abuser of the just because fallacy. As for the mother's reason, she is making a claim that can be factual or not. Even though this is beyond the scope of this site, I think we can safely say that her claim is true (i.e., let's assume causality). To be true, there just needs to be two cases of people dying of heart problems as a direct (proven) result of the energy drinks. People die of all kinds of weird things every day, including drinking too much water, so this is not at all unbelievable.
The mother COULD BE making an well-reasoned evaluative judgement based on the information she has, which might be factual or might not be. Here are the factors that she should be considering:
What is proven risk of drinking these (in terms of percentages)?
How does this risk relate to other things she lets the son do?
How solid is the research?
Has this been adequately researched?
What would the son lose by NOT drinking these?
Basically, we are talking about a cost/benefit analysis based on actual data rather than cognitive biases (such as the availability heuristic). The problem is, her reason given does not reflect this. She is focused only on the potential cost.
Now for the son:
Son: Well, you are more likely to get killed by your spouse than by energy drinks, so by your logic, I shouldn’t be getting married!
Same as with the mother, the claim "Well, you are more likely to get killed by your spouse than by energy drinks" is either factually true or not. Since we gave the mother the benefit of the doubt, let's do the same for the son and assume this is factually true (it sure seems like it would be, but "seems" is not always good enough). Now, in the second part, the son is drawing a conclusion from the premises, although some might be implied. Let's expand his argument (some assumptions being made here):
P1. Activity X kills Y percent of people.
P2. Y percent it too high of a risk to justify activity X.
P3. Activity Z is even more risky than Activity X.
C. Therefore, activity Z it too risky to be justified.
The above looks good, but it exposes a flaw in the mother's reason given. Recall that the mother gave a reason that only included the risk and not the benefit. The son essentially use a reductio ad absurdum to expose the flaw in the mother's argument. Risk is only one side of the equation.
In summary, I wouldn't call "fallacy" on either the mother or the son, but I would say that the mother has made a weak argument.