"They will call me if she dies during the night."
Translation:De ringer mig om hon dör under natten.
from my experience, it's because words like that (at, to, during, for) honestly just are sometimes memorized in languages. It's like that in english (for example, you get IN a car, but ON a bus) and it's like that in spanish, especially with many reflexive verbs. Unfortunately, we sometimes just have to memorize these things.
De kommer att ringa mig om hon dör under natten is an accepted answer already, and it's a good one. I'm not sure about De ska ringa mig … though. I don't think its' a very good translation in this case, since it adds something that isn't there in the English sentence. In the ska future, there's always a sense that somebody wants or intends for something to happen, it's not neutral. If you imagine that the context of the sentence is that "they" promised you this, then it could work. Then they are the ones who have the intention. The interesting thing about ska is that there's an intention there, but the intention could come either from the speaker or from the person s/he is talking about.
Anyway we should probably approve this version, especially since that context is indeed very likely, but keep in mind that ska can be a little tricky this way, it can sometimes make it sound like you're giving out orders.