If I cast Stonecoil Serpent with X = 0, it starts with 0/0, and immediately resolves so it dies?
Is there a way to save it?
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Yes, you can save the Serpent, but it must be set up ahead of time. There are two main ways of saving it that I can think of.
Either of these will make sure the Serpent has toughness greater than 0 by the time state-based actions are checked. Note that this (in particular point 2) is in contrast to triggered abilities (like The Great Henge), which are too late to save the serpent.
Between the time that the Stonecoil Serpent spell actually resolves and when it is put into the graveyard, no player gets priority to do anything. This is because state-based actions are checked when a player is about to get priority. So any method of saving it would have to be done either before the servant resolves (or before it is cast), or after it has already entered the graveyard.
For example, if you had a Always Watching in play before you cast the serpent, then that would save it. Or you could return it from the graveyard to your hand with various cards. If you just returned it to the battlefield from the graveyard, it would just die again immediately, unless something else had changed that matters.
Arthur's answer pretty much covers everything you need to know. I'd only add that, in a Silver Bordered game, you can also use Rules Lawyer to save a 0 toughness creature.