Question: What kind of fallacy would this be?

Let's see!
When the mother says "There have been cases of people dying of heart problems after drinking them!", she can be guilty of the questionable cause fallacy (https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/148/Questionable-Cause) or the oversimplified cause fallacy. (https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/91/Oversimplified-Cause-Fallacy). The mere fact that some fraction of people died after drinking energy drinks shows a correlational link, not necessarily a causal link. Also, if it is only the case for "cases of people" and not the majority of people, it could be the case that energy drinks are only a contributing factor combined with other factors (those people's heart health, other health conditions that are incompatible with drinking energy drinks, etc) that cause death, those other factors being highly unlikely to be available in her son. Also, since the majority of healthy people don't die after drinking an energy drink, she can be guilty of an appeal to possibility fallacy (https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/41/Appeal-to-Possibility) where she's demanding her son's choices to be based on possibilities, not high likelihoods and probabilities.
Now for the son! His response "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!" can be seen as a good response that attempts to show the irrational fear of the mother and the absurd conclusions her logic may lead to, but the mother can say that her son is committing a false equivalence fallacy (https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/245/False-Equivalence) because marriage (in the mother's opinion) is a necessary action to take and can yield many benefits to the person that outweigh the small likelihood of the person getting killed by their spouse, whereas energy drinks can be completely avoided without any advantages to miss from not drinking them.